Download Exponential distribution

UNIT-1
Random variables:
Random variables:
Discrete
Continuous
Probability
Distribution
Discrete
Binomial D.
Continuous
Poisson D.
Related Problems, mean,
Normal D.
Variance, standard deviation
In a random experiment, the outcomes are governed by
chance mechanism and the sample space’s consists of all
outcomes of the experiment when the elements of the
sample space are non-numeric, they can be quantified by
assigning a real no to every event of the sample space. This
assignment rule is known as random variable.
Random Variable:
A random variable X on a sample space S is a function
X:S→R from S to the set of real no.s which assigns a real
number X(x) to each sample point of S.
s
r
x
x(x)
[i.e., the pre image of every element of R is an event of S]
Range:
The range space Rx is the set of all possible values of
X:Rx≤R
Note: Although X is called a random variable, it is intact a
single valued fn.
X denotes random variable and x denotes one of its values.
Discrete Random Variable:
A random variable X is said to be discrete random variable
if its set of all possible outcomes, the sample space S is
countable (termite or countable infinite).
Counting Problems give rise to discrete random variables.
Continuous Random variable:
A random variable X is said to be continuous random
variable if S contains infinite no.s equal to the no. of points
on a live segment.
i.e., it takes all the possible values in an interval.
(An internal contains uncountable no. of possible values).
Ex: Consider the experiment of tossing a coin twice.
Sample space S={HH,HT,TH,TT}. Define X:S→R by
X(s)=no. of heads in S
. X (s )  2 X (s )  1X (s )  1X (s )  0
Range of X   X (8) :8  s  0,1, 2
1
2
3
4
2. Consider the experiment of throwing a pair of dice and
noting sum
S={(1,1)(1,2)_ _ _ _(6,6)}
Then the random variable for this experiment is defined as
X : S  RbyX (i, j )  i  j(i, j )  S .
{1.If X and Y are two random variables defined on S and a
and b are two real no.s, then.
(i)ax+by is also a random variable . In particular x-y is also
a random variable.
(ii) xy is also a random variable
(iii) IfX (s)  0s  sthen 1/x is also a random a
variable.}
.
It an a random experiment, the event corresponding to
a number a occurs, then the corresponding random variable
X assumes a and the probability of that event is denoted by
p(x=a) similarly, the probability of the event x assuming
any value in the interval a<x<b).
The probability of the event x≤c is written as p(x.≤c).
Note that more than one random variable can be
defined in a sample space
Discrete random variable:
It we can count the possibilities, of x1 then x is discrete.
Ex: The random variable x=the sum of the dots on two dice
is discrete.
X can assume the value 2,3,4,5………12.
Continuous:
In an interval of real no.s, there are an infinite no. of
possible values.
Ex: The random variable x= the time that an athlete crosses
the winning live.
[Probability density function:
The p.d.f. of a random variable x denoted by of (x) has the
favoring properties.
f ( x) 
1.
 f ( x)dx  1
2.


3.
P( E )   f ( x)dx
Where E is any event.
E
Note: P(F) = 0 does not imply that E is nill event or
impossible event.]
Probability distribution or distribution:
The probability distribution or distribution f(x) of a random
variable X is a description of the set of possible values of
(range of x) along with the probability associated with each
of x
Ex: Let x=the no.of heads in tossing two coins
X=x
0
1
2
F(x) = P(X=x)
¼
2/4=1/2
1/4
Cumulative distribution function:
The cumulative distribution function for a random variable
x is defined by F(x) = P(x≤x) where x is any real no. (…….)
Properties:
1. If a<b then p(a<x≤b) = F(b)-F(a).
2. P(a≤x≤b) = p(x=a)+f(b)-f(a)
3. P(a<x<b) = F(b)-F(a)-p(x=b)
4. P(a≤x<b)=F(b)-F(a)-P(x)
According to the type of random variables, we have two
types of probability distributions.
1 Discrete probability distribution
2. Continuous probability distribution.
Discrete Probability distribution:
Let x be a discrete random variable. The discrete
probability distribution function f(x) for x is given by
satisfying the properties
f ( x)  p( x  x)xorf ( xi )  p( x  xi )i  1, 2...
1. P(xi) ≥0 V i
2. ∑P(xi) = 1
1. f(x)>,0 V x
2. ∑f(x) = 1 x←x
The discrete probability function can also be called as
probability mass function.
Any function satisfying the above 2 properties, will be a
discrete probability fn. or probability mass function.
X=x x1 x2 x3
P(x=x) P1 P2 P3
Ex: x: the sum of no.s which turn on tossing a pair of dice.
X=xi
2
3
4
5
6
7
8
9
10 11
12
P(x=xi)1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36
1/36
1 .P(xi)≥0
2. ∑P(xi)=1
X x1 x2 ---------------Xn
P(x) p(x1)
p(x2)
--------------P(Xn)
1. P (X<xi) = P (x1) + P (X2) + ----------+P(Xi-i)
2. P (X≤Xi) = P (x1) + P (X2)+ ---------+P (Xi-1) +
P(Xi)
3. P (X>Xi) = 1 – P (x≤xi)
Check whether the following can act as discrete probability
functions.
1. F(x) = X - 2/2 for X = 1,2,3,4
2. F(x) =XF2/25 x = 0,1,2,3,4
(1)
cannot f(x)<0 for x=1
(2)
cannot ∑f(x)≠1
General properties:
Expectation, mean, variance & Standard deviation :
Expectation:
Let a random variable x assumes the values x1,x2--------xn
with respective probabilities f1,f2,--------fn. Then the
expectation of X E(x) is defined as the sum of different
values of x and the corresponding probabilities.
E( X ) 
n

i 1
pi xi
Results:
1. It x is a random variable and k is a constant then
a. F(x+k) = F(x)+k
b. E(x)
2 If x and y are two discrete random variables, then E(x+y)
= E(x)+E(y)
Note:
1. E(x+y+z) =E(x) + E(y) + E(z)
2. E(ax+by) = a E(x) + bE(y)
3. E(xyz) = E(x) E(y) E(z)
Mean:
The mean value u of a discrete distribution
function is given by
n
px
i 1
i
i
 E( X )
Variance: The variance of a discrete distribution function is
given by
n
   pi x 2i   2
2
i 1
Standard deviation:
It is nothing but the positive square root of the variance.
  p x 
n
2
2
i 1
i
2
i
Continuous Probability distribution:
When a random variable x takes every value in an interval,
it is called an continuous random variable.
Ex: Temperature, heights and weights.
The continuous random variable will form a curve that can
be used to calculate the probabilities via areas.
Def: Let x be a continuous random variable. A function f(x)
is said to be a continuous probability function of x if for
any [a,b] a,b←R such that
1. f(x) ≥0
2.  f ( x)dx  1

The total area bounded by

graph f(x) and horizontal axis is
3. p(a≤x≤b) =  f (t )dt
a and b any two values of x
b
a
satisfying a<b.
Continuous probability function f(x) is also called as
probability density function.
Def: the cumulative distribution function of a continuous
random variable x is defined by f(x) = p (x≤xx)
x
F(x) = 
f (t )dt
where f(x) is Continuous probability function

By def. f(x) =
d
( F ( x ))
dx
General Properties;
Let X be a random variable with p.d.f f(x)
The mean of X,

   xf ( x)dx

The variance of X denoted by


   ( x   ) f ( x)dx   x2 f ( x)dx   2
2
2


The standard deviation of X denoted by


  x   
2
f ( x)dx
Results:
If X is a continuous random variable and Y=ax+b then
E(Y)=aE(X)+b
And V(X)= a Var(Y)
Var(X+k)=Var(X)
2
V(kX)=



Median:
k 2 Var(X)
xf ( x)dx
In case of continuous distribution, Mediam is a point which
divides the entire distribution into two equal parts.
If X is defined from a to b,M is median then
M
b
1
f
(
x
)
dx

f
(
x
)
dx

a
M
2
Mode:
Mode is the value of x for which f(x) is maxium.
Mode is given by
f ( x)  0 f ( x)  0 fora  x  b
CHEBYSHEV’S INEQUALITY
: "The probability that the outcome of an experiment with the random variable
will fall more than standard deviations beyond the mean of , , is less than
."
Or: "The proportion of the total area under the pdf of outside of standard
deviations from the mean is at most
."
Proof
Let be the sample space for a random variable, , and let
stand for the
pdf of . Let , and partition , such that for every sample point in
Then
.
Clearly
Since the term that evaluates to the variance in
subtracted on the right-hand side.
has been
For any sample point in
Notice that the direction of the inequality changes since
squaring causes the right-hand expression to become
positive.
And for any sample point in
So, for any sample point in
, and so
or
, it can be said that
Dividing each side of the inequality by
Or, in other terms
results in
UNIT –II
Binomial distribution:
Binomial and Poisson distributions are related to
discrete random variables and normal distribution is related
to continuous random variables.
In many cores, it is desirable to have situations called
repeated trials. For this, we develop a model that is useful
in representing the probability distributions pertaining to
the no. of occurrence of an event in repeated trials of an
experiment.
Binomial distribution is discovered by James
Bernoulli
Bernoulli Trials:
If there are n trials of an experiment in which each
trial has only two mutually exclusive, equally likely and
independent outcomes, then they are called Bernoulli trials,
Let us denote the two outcomes by success and failure.
The Bernoulli distribution is a discrete distribution
having two possible outcomes labeled by
and
in
which
("success") occurs with probability and
("failure") occurs with probability
, where
. It
therefore has probability density function
which can also be written
the binomial distribution
The Binomial Distribution is one of the discrete
probability distribution. It is used when there are
exactly two mutually exclusive outcomes of a trial.
These outcomes are appropriately labeled Success
and Failure. The Binomial Distribution is used to
obtain the probability of observing r successes in n
trials, with the probability of success on a single trial
denoted by p.
Formula:
P(X = r) = nCr p r (1-p)n-r
where,
n = Number of events.
r = Number of successful events.
p = Probability of success on a single trial.
nCr = ( n! / (n-r)! ) / r!
1-p = Probability of failure.
Example: Toss a coin for 12 times. What is the
probability of getting exactly 7 heads.
Step 1: Here,
Number of trials n = 12
Number of success r = 7 (since we define
getting a head as success)
Probability of success on any single trial p
= 0.5
Example: Toss a coin for 12 times. What is the
probability of getting exactly 7 heads.
Step 1: Here,
Number of trials n = 12
Number of success r = 7 (since we define
getting a head as success)
Probability of success on any single trial p
= 0.5
Step 2: To Calculate nCr formula is used.
nCr = ( n! / (n-r)! ) / r!
= ( 12! / (12-7)! ) / 7!
= ( 12! / 5! ) / 7!
= ( 479001600 / 120 ) / 5040
= ( 3991680 / 5040 )
= 792
Step 3: Find pr.
pr = 0.57
= 0.0078125
Step 4: To Find (1-p)n-r Calculate 1-p and n-r.
1-p = 1-0.5 = 0.5
n-r = 12-7 = 5
Step 5: Find (1-p)n-r.
= 0.55 = 0.03125
Step 6: Solve P(X = r) = nCr p r (1-p)n-r
= 792 × 0.0078125 × 0.03125
= 0.193359375
The probability of getting exactly 7 heads is 0.19
Step 2: To Calculate nCr formula is used.
nCr = ( n! / (n-r)! ) / r!
= ( 12! / (12-7)! ) / 7!
= ( 12! / 5! ) / 7!
= ( 479001600 / 120 ) / 5040
= ( 3991680 / 5040 )
= 792
Step 3: Find pr.
pr = 0.57
= 0.0078125
Step 4: To Find (1-p)n-r Calculate 1-p and n-r.
1-p = 1-0.5 = 0.5
n-r = 12-7 = 5
Step 5: Find (1-p)n-r.
= 0.55 = 0.03125
Step 6: Solve P(X = r) = nCr p r (1-p)n-r
= 792 × 0.0078125 × 0.03125
= 0.193359375
The probability of getting exactly 7 heads is 0.19
Mean and variance
If X ~ B(n, p) (that is, X is a binomially distributed random
variable), then the expected value of X is
and the variance is
This fact is easily proven as follows. Suppose first that we have a
single Bernoulli trial. There are two possible outcomes: 1 and 0,
the first occurring with probability p and the second having
probability 1 − p. The expected value in this trial will be equal to μ
= 1 · p + 0 · (1−p) = p. The variance in this trial is calculated
similarly: σ2 = (1−p)2·p + (0−p)2·(1−p) = p(1 − p).
The generic binomial distribution is a sum of n independent
Bernoulli trials. The mean and the variance of such distributions
are equal to the sums of means and variances of each individual
trial:
Mode and median
Usually the mode of a binomial B(n, p) distribution is equal to
⌊(n + 1)p⌋, where ⌊ ⌋ is the floor function. However when
(n + 1)p is an integer and p is neither 0 nor 1, then the distribution
has two modes: (n + 1)p and (n + 1)p − 1. When p is equal to 0 or
1, the mode will be 0 and n correspondingly. These cases can be
summarized as follows:
In general, there is no single formula to find the median for a
binomial distribution, and it may even be non-unique. However
several special results have been established:

If np is an integer, then the mean, median, and mode
coincide.
Any median m must lie within the interval ⌊np⌋ ≤ m ≤ ⌈np⌉.
POISSON DISTRIBUTION
In probability theory and statistics, the Poisson
distribution is a discrete probability distribution
that expresses the probability of a number of events
occurring in a fixed period of time if these events
occur with a known average rate and independently
of the time since the last event. The Poisson
distribution can also be used for the number of
events in other specified intervals such as distance,
area or volume.
The distribution was discovered by Siméon-Denis
Poisson (1781–1840) and published, together with
his probability theory, in 1838 in his work
Recherches sur la probabilité des jugements en
matières criminelles et matière civile ("Research on
the Probability of Judgments in Criminal and Civil
Matters"). The work focused on certain random
variables N that count, among other things, a
number of discrete occurrences (sometimes called
"arrivals") that take place during a time-interval of
given length. If the expected number of occurrences
in this interval is λ, then the probability that there
are exactly k occurrences (k being a non-negative
integer, k = 0, 1, 2, ...) is equal to
where

e is the base of the natural logarithm (e =
2.71828...)


k is the number of occurrences of an event - the
probability of which is given by the function
k! is the factorial of k
λ is a positive real number, equal to the expected
number of occurrences that occur during the given
interval. For instance, if the events occur on average
4 times per minute, and you are interested in the
number of events occurring in a 10 minute interval,
you would use as model a Poisson distribution with

λ = 10*4 = 40.
As a function of k, this is the probability mass
function. The Poisson distribution can be derived as a
limiting case of the binomial distribution.
The Poisson distribution can be applied to systems
with a large number of possible events, each of
which is rare. A classic example is the nuclear decay
of atoms.
The Poisson distribution is sometimes called a Poissonian,
analogous to the term Gaussian
The parameter λ is not only the mean number of
occurrences , but also its variance . Thus, the
number of observed occurrences fluctuates about its
mean λ with a standard deviation
Datat sometimes arise as the number of occurrences ( ) of an
event per unit time or space, e.g., the number of yeast cells per cm2
on a microscope slide. Under certain conditions (see below), the
random variable is said to follow a Poisson distribution, which,
as a count, is type of discrete distribution. Occurrences are
sometimes called arrivals when they take place in a fixed time
interval.
The Poisson distribution was discovered in 1838 by Simeon-Denis
Poisson as an approximation to the binomial distribution, when the
probability of success is small and the number of trials is large.
The Poisson distribution is called the law of small numbers
because Poisson events occur rarely even though there are many
opportunities for these evens to occur.
Poisson Experiment
The number of occurrences of an event per unit time or space will
have a Poisson distribution if:



the rate of occurrence is constant over time or space;
past occurrences do not influence the likelihood of future
occurrences;
simultaneous occurrences are nearly impossible.
Poisson Distribution
In the binomial distribution the probability P(r) of r
successes is given by P(r)=
e  r
x!
. This is known as Poisson
distribution.
Put r=0,1,2,……etc., the probabilities of 0,1,2,…..successes
 2 
 r 
are given by e ,  e , e ,............ e
2
r!


# The sum of these probabilities is unity.
A random variable X is said to follow a Poisson distribution if
it assumes only nonnegative values and its probability is given

 
e
by p(x,  ) =P(X=x)=
x!
x
;x=0,1,2,…….
O , otherwise.
^ Poisson Moments
The mean of a Poisson random variable is , i.e.,
. The
mean is a rate, e.g., a temporal rate for time events. For example, if
0.5 phone calls per hour are received on a home phone during the
day, then the mean number of phone calls between 9 A.M. and 5
P.M. is
.
The Poisson has the interesting property that the variance is also ,
i.e.,
. Thus, unlike the normal distribution, the
variance of a Poisson random variable depends on the mean.
The standard deviation is given by   
The mode for this distribution is(1)  1and if  is an intezer.
(2)integral part of  if  is not
an intezer.
Poisson Approximation to the Binomial
Consider a binomial distribution consisting of trials with
probability of success
, i.e,.
for
. If is sufficiently
large, then the binomial probability:
,
i.e., the binomial probability is approximately equal to the
corresponding Poisson probability. Note that
, the mean of
the binomial distribution.
Thus, binomial probabilities, which are hard to compute for large
, can be approximated by corresponding Poisson probabilies. For
example, suppose 10,000 soldiers are screened for a rare blood
disease (
). We want the probability at least 10 soldiers test
positive for the disease, i.e, we want
, where
. This is difficult to compute using the
binomial distribution, but much easier for the Poisson with
.
Related distributions.


If and are independent, and Y = X1 + X2, then
the distribution of X1 conditional on Y = y is a
binomial. Specifically, . More generally, if X1,
X2,..., Xn are Poisson random variables with
parameters λ1, λ2,..., λn then
 The Poisson distribution can be derived as a
limiting case to the binomial distribution as the
number of trials goes to infinity and the
expected number of successes remains fixed.
Therefore it can be used as an approximation of
the binomial distribution if n is sufficiently large
and p is sufficiently small. There is a rule of
thumb stating that the Poisson distribution is a
good approximation of the binomial distribution
if n is at least 20 and p is smaller than or equal
to 0.05. According to this rule the approximation
is excellent if n ≥ 100 and np ≤ 10.For
sufficiently large values of λ, (say λ>1000), the
normal distribution with mean λ, and variance λ,
is an excellent approximation to the Poisson
distribution. If λ is greater than about 10, then
the normal distribution is a good approximation
if an appropriate continuity correction is
performed, i.e., P(X ≤ x), where (lower-case) x
is a non-negative integer, is replaced by
P(X ≤ x + 0.5).
If the number of arrivals in a given time follows
the Poisson distribution, with mean = λ, then
the lengths of the inter-arrival times follow the
Exponential distribution, with rate 1 / λ.
Occurrence
The Poisson distribution arises in connection with
Poisson processes. It applies to various phenomena
of discrete nature (that is, those that may happen 0,
1, 2, 3, ... times during a given period of time or in a
given area) whenever the probability of the
phenomenon happening is constant in time or space.
Examples of events that may be modelled as a
Poisson distribution include:
The number of soldiers killed by horse-kicks
each year in each corps in the Prussian cavalry.
This example was made famous by a book of
Ladislaus Josephovich Bortkiewicz (1868–1931).
 The number of phone calls at a call center per
minute.
 The number of times a web server is accessed
per minute.
 The number of mutations in a given stretch of
DNA after a certain amount of radiation.

[Note: the intervals between successive Poisson
events are reciprocally-related, following the
Exponential distribution. For example, the lifetime of
a lightbulb, or waiting time between buses.]
The law of rare events
In several of the above examples—for example, the
number of mutations in a given sequence of DNA—
the events being counted are actually the
outcomes of discrete trials, and would more
precisely be modelled using the binomial
distribution. However, the binomial distribution
with parameters n and λ/n, i.e., the probability
distribution of the number of successes in n
trials, with probability λ/n of success on each
trial, approaches the Poisson distribution with
expected value λ as n approaches infinity. This
provides a means by which to approximate
random variables using the Poisson distribution
rather than the more-cumbersome binomial
distribution.
This limit is sometimes known as the law of
rare events, since each of the individual
Bernoulli events each rarely trigger. The name
may be misleading because the total count of
success events in a Poisson process need not
be rare if the parameter λ is not small. For
example, the number of telephone calls to a
busy switchboard in one hour follows a Poisson
distribution with the events appearing frequent
to the operator, but they are rare from the
point of the average member of the population
who is very unlikely to make a call to that
switchboard in that hour.
The "law of small numbers"

The word law is sometimes used as a synonym
of probability distribution, and convergence in
law means convergence in distribution.
Accordingly, the Poisson distribution is
sometimes called the law of small numbers
because it is the probability distribution of the
number of occurrences of an event that happens
rarely but has very many opportunities to
happen
Examples for rare events:
1. No. of printing mistakes per page.
2. No. of accidents on highway.
3. No. of bad cheques at a bank.
4. No. of blind persons
5. No. of noble prize winners
6. No. of Bharath rattans.
Normal Probability Distribution
Normal distribution:
Normal distribution is one of the most widely used
continuous probability distribution in applications of
statistical methods. It is of tremendous importance in the
analysis and evaluation of every aspect of experimental
date in science and medicine.
Def: Normal distribution is the probability distribution of a
continuous random variable x, known as normal random
variables or normal variate.
It is also called Gaussian distribution.
In this the variate can take all values with in a given
range.Examples are measurements (e.g., weight, time, etc.) can be
well approximated by the normal distribution.
It is another limiting form of binomial distribution for large values
of n when neither p nor q is very small. The normal distribution is
derived from the binomial distribution
By increasing the number of trials indefinitely.
A randon variable is said to be a normal distribution ,if its
probability distribution is given by
f ( x; , ) 
1
e
 2

( x   )2
2 2
  x  ,     ,  0
Where  =Mean and  =Standard deviation are two parameters
of the normal distribution.
Mean of Normal distribution
The mean of the normal distribution with b,  as parameters is b.
Variance of normal distribution is  2
The standard deviation of the normal distribution is

Mode
Mode is the value of x for which f(x) is maximum.i.e;the mode
is the solution of f(x)=0 and f  ( x)  0
f ( x)  
x
. f ( x)

f ( x)  
f ( x)  0  x    0  x  
1
0
2
 2
Hence x=  is the mode.
Median
M
We know that

1
2

f ( x)dx 

Using this we have Median =
Hence for the Normal Distribution Mean = Mode = Median.
Hence the distribution is symmetrical.
Chief characteristics:
1. The graph of the normal distribution y=f(x) in x-y plane
is known as the normal curve.
2. The curve is bell shaped and symmetrical about the line
x= 
3. Area under the normal curve is unity i.e., it represents
total population.
4. Mean = median =mode.
5.x- axis is an asymptote to the curve and the points of
inflexion of the curve at x=   
7. Since mean =  , the line x =  -- divides the total area
into two equal parts.
8. No portion of the curve lies below x – axis.
9.The prob. that the normal vaiate x with mean  – and
c.d. between    and    is 68.26%
10.Area of the normal curve between   2 and   2
is95.44%
11. Area of the normal curve between   3 and   3
is99.73%
Def:
The normal distribution for   0 AND  1
known as STANDARD NORMAL DISTRIBUTION.
is
Importance and applications of normal distribution:
Normal distribution plays a very important role in statistical
theory because of the following reasons
1. Most of the distributions for example, binomial, Poisson
etc can be approximated by Normal distribution..
2. Since it is a limiting case of Binomial distribution for
exceptionally large numbers, it is applicable to many
applied problems in kinetic theory of gases and fluctuations
in the magnitude of an electric current.
3. It is variable is normally distributed, it can sometimes be
brought to normal form by simple transformation of the variable.
4. The proofs of all the tests of significance in sampling are based
on the fundamental assumption that the population from which the
samples have been drawn are normal.
5. Normal distribution finds large applications in statistical quality
Control.
6. Many of the distributions of sample statistic, the distributions of
sample mean, sample variance etc tend to normality for large
samples and as such they can best be studied with the help of
normal curve.
Area property:
x
By taking z 

the standard normal curve is formed.
The probability that the normal variate x with mean – and s.d.
lies, between two specific values x1 and x2 with x1  x2 , can be
obtained using area under the standard normal curve as follows.
1.Put
2.To find
(i) If
x
z
, forx1 , x2 , findz1 , z2

P( x1  x  x2 )  P( z1  z  z2 )
z1 , z2
are positive or negative then
P( z1  z  z2 )  A( z2 )  A(z1)
(ii)If
z1  oand , z2  o
P( x1  x  x2 )
 A( z2 )  A( z1 )
P( z  z1 )
if , z1  0, then
P( z  z1 )  0.5  A( z1 )
if , z1  0, then
P( z  z1 )  0.5  A( z1 )
then
(4)
To find
P( z  z1 )
if , z1  0, then
P( z  z1 )  0.5  A( z1 )
if , z1  0, then
P( z  z1 )  0.5  A( z1 )
Def: A normal random variable with mean   0 and variance  2  1 is called standard normal variable. Its probability density
function is given by
1 z2/2
 ( s) 
e   z  
2
Def: The cumulative distribution fn. of a standard normal random
variable is
1
F ( z )  p( Z  z ) 
2

 z /2
 e dz
2

Normal Distribution as Approximation to Binomial
Distribution
Normal Distribution provides a valuable approximation to
Binomial when the sample sizes are large and the probability of
successes and failures is not close to zero.
Poisson Approximation to Binomial Distribution
Poisson provides an approximation to Binomial Distribution when
the sample sizes are large and the probability of successes or
failures is close to zero.
Normal Approximation to Poisson
The Poisson can be approximated fairly well by Normal
Distribution when λ is large.
Normal approximation is binomial distribution:
When n is very large, it is very difficult to calculate the
probabilities by using binominal distribution. Normal distribution
is a limiting case of binomial distribution under the following
conditions.
(1) N, the no. of trials very large n --(2) neither p nor q is very small.
For a b.d. E(x) = np var(x) = npq.
Then the standard normal variate
Z
x u


x  np
npq
tends to the distribution of standard normal variable
given by
z
1
( z ) 
e 2   z  
2
2
If p  q , and for large n, we can approximate binomial curve by
normal curve. Here the interval becomes (x-1/2,x+1/2)
Note: If X is a poisson variable with mean  then the standard
normal variable Z= x   and the probability can be calculated as

explained above
Poisson distribution approaches the normal distribution as   
A uniform distribution, sometimes also known as a
rectangular distribution, is a distribution that has constant
probability.
The probability density function and cumulative distribution
function for a continuous uniform distribution on the interval
are
(1)
(2)
These can be written in terms of the Heaviside step function
as
(3)
(4)
the latter of which simplifies to the expected
.
for
The continuous distribution is implemented as
UniformDistribution[a, b].
For a continuous uniform distribution, the characteristic
function is
(5)
If
and
, the characteristic function simplifies to
(6)
(7)
The moment-generating function is
(8)
(9)
(10)
and
(11)
(12)
The moment-generating function is not differentiable at zero,
but the moments can be calculated by differentiating and
then taking
. The raw moments are given analytically by
(13)
(14)
(15)
The first few are therefore given explicitly by
(16)
(17)
(18)
(19)
The central moments are given analytically by
(20)
(21)
(22)
The first few are therefore given explicitly by
(23)
(24)
(25)
(26)
The mean,
variance,
skewness, and
kurtosis excess
are therefore
Exponential
distribution
Inion (a.k.a. negative
exponential
distribution) is a
family of continuous
probability
distributions. It
describes the tim
between events in a
Poisson process, i.e.
a process in which
events occur
continuously and
independently at a
constant average rate.
Note that the exponential distribution is not the same as the class of
exponential families of distributions, which is a large class of
probability distributions that includes the exponential distribution
as one of its members, but also includes the normal distribution,
binomial distribution, Poisson, and many others.

Characterization
Probability density function
The probability density function (pdf) of an exponential
distribution is
Here λ > 0 is the parameter of the distribution, often called the rate
parameter. The distribution is supported on the interval [0, ∞). If a
random variable X has this distribution, we write X ~ Exp(λ).
A commonly used alternative parameterization is to define the
probability density function (pdf) of an exponential distribution as
where β > 0 is a scale parameter of the distribution and is the
reciprocal of the rate parameter, λ, defined above. In this
specification, β is a survival parameter in the sense that if a
random variable X is the duration of time that a given biological or
mechanical system manages to survive and X ~ Exponential(β)
then E[X] = β. That is to say, the expected duration of survival of
the system is β units of time. The parameterisation involving the
"rate" parameter arises in the context of events arriving at a rate λ,
when the time between events (which might be modelled using an
exponential distribution) has a mean of β = λ−1.
The alternative specification is sometimes more convenient than
the one given above, and some authors will use it as a standard
definition. This alternative specification is not used here.
Unfortunately this gives rise to a notational ambiguity. In general,
the reader must check which of these two specifications is being
used if an author writes "X ~ Exponential(λ)", since either the
notation in the previous (using λ) or the notation in this section
(here, using β to avoid confusion) could be intended.
Occurrence and applications
The exponential distribution occurs naturally when describing the
lengths of the inter-arrival times in a homogeneous Poisson
process.
The exponential distribution may be viewed as a continuous
counterpart of the geometric distribution, which describes the
number of Bernoulli trials necessary for a discrete process to
change state. In contrast, the exponential distribution describes the
time for a continuous process to change state.
In real-world scenarios, the assumption of a constant rate (or
probability per unit time) is rarely satisfied. For example, the rate
of incoming phone calls differs according to the time of day. But if
we focus on a time interval during which the rate is roughly
constant, such as from 2 to 4 p.m. during work days, the
exponential distribution can be used as a good approximate model
for the time until the next phone call arrives. Similar caveats apply
to the following examples which yield approximately
exponentially distributed variables:



The time until a radioactive particle decays, or the time
between clicks of a geiger counter
The time it takes before your next telephone call
The time until default (on payment to company debt holders)
in reduced form credit risk modeling
Exponential variables can also be used to model situations where
certain events occur with a constant probability per unit length,
such as the distance between mutations on a DNA strand, or
between roadkills on a given road.
Properties
Mean, variance, and median
The mean or expected value of an exponentially distributed
random variable X with rate parameter λ is given by
In light of the examples given above, this makes sense: if you
receive phone calls at an average rate of 2 per hour, then you can
expect to wait half an hour for every call.
The variance of X is given by
The median of X is given by
where ln refers to the natural logarithm. Thus the absolute
difference between the mean and median is.
Independent events
The standard definition says:
Two events A and B are independent if and only if Pr(A ∩ B)
= Pr(A)Pr(B).
Here A ∩ B is the intersection of A and B, that is, it is the event that
both events A and B occur.
More generally, any collection of events—possibly more than just
two of them—are mutually independent if and only if for every
finite subset A1, ..., An of the collection we have
variables X and Y are independent if and only if for every a and b,
the events {X ≤ a} and {Y ≤ b} are independent events as defined
above. Mathematically, this can be described as follows:
The random variables X and Y with distribution functions FX(x)
and FY(y), and probability densities ƒX(x) and ƒY(y), are
independent if and only if the combined random variable (X, Y) has
a joint cumulative distribution function
or equivalently, a joint density
.
Similar expressions characterise independence more generally for
more than two random variables.
An arbitrary collection of random variables – possibly more than
just two of them — is independent precisely if for any finite
collection X1, ..., Xn and any finite set of numbers a1, ..., an, the
events {X1 ≤ a1}, ..., {Xn ≤ an} are independent events as defined
above.
If any two of a collection of random variables are independent,
they may nonetheless fail to be mutually independent; this is called
pairwise independence.
If X and Y are independent, then the expectation operator E has the
property
and for the variance we have
so the covariance cov(X, Y) is zero. (The converse of these, i.e. the
proposition that if two random variables have a covariance of 0
they must be independent, is not true. See uncorrelated.)
Two independent random variables X and Y have the property that
the characteristic function of their sum is the product of their
marginal characteristic functions:
but the reverse implication is not trues
Conditionally independent random variables
Conditional independence
Intuitively, two random variables X and Y are conditionally
independent given Z if, once Z is known, the value of Y does not
add any additional information about X. For instance, two
measurements X and Y of the same underlying quantity Z are not
independent, but they are conditionally independent given Z
(unless the errors in the two measurements are somehow
connected).
The formal definition of conditional independence is based on the
idea of conditional distributions. If X, Y, and Z are discrete random
variables, then we define X and Y to be conditionally independent
given Z if
for all x, y and z such that P(Z = z) > 0. On the other hand, if the
random variables are continuous and have a joint probability
density function p, then X and Y are conditionally independent
given Z if
for all real numbers x, y and z such that pZ(z) > 0.
If X and Y are conditionally independent given Z, then
for any x, y and z with P(Z = z) > 0. That is, the conditional
distribution for X given Y and Z is the same as that given Z alone.
A similar equation holds for the conditional probability density
functions in the continuous case.
Independence can be seen as a special kind of conditional
independence, since probability can be seen as a kind of
conditional probability given no events.