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Discrete and Continuous Distributions
Paper: Probability and Statistics
Lesson: Discrete and Continuous Distributions
Lesson Developer: Nikhil Khanna
College/Department: Research Scholar,
Department of Mathematics,
University of Delhi
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Discrete and Continuous Distributions
Contents
1.
Introduction ................................................................................. 3
2.
Discrete Distributions .................................................................... 3
Discrete Uniform Distribution ............................................................... 3
Binomial Distribution and Bernoulli distribution ...................................... 4
Poisson Distribution ............................................................................ 7
Geometric Distribution ...................................................................... 10
Negative Binomial Distribution ........................................................... 12
Some Solved Problems ..................................................................... 15
3. Continuous Distributions .................................................................. 17
Uniform Distribution ......................................................................... 17
Exponential Distribution .................................................................... 19
Normal Distribution .......................................................................... 21
Exercises ........................................................................................... 26
References ........................................................................................ 27
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Discrete and Continuous Distributions
1. Introduction
Till now, we have studied the concepts of random or non-deterministic experiments,
classical definition of Probability function and the axiomatic approach to the Probability
Theory. Further, discrete distributions, probability mass functions, continuous distributions
and probability density functions have also been studied.
In the present chapter, we will study some of the probability distributions which have
extremely important applications in the field of statistical theory. Further, we shall study
their parameters, i.e., the quantities that are fixed for one distributions but changes or
takes different values for different members of families of distributions of the same kind.
The most common parameters are the lower moments, mainly mean
and variance
.
Various forms of discrete distributions such as uniform, binomial, Poisson, geometric,
negative binomial and continuous distributions such as uniform, normal, exponential will be
studied with the help of various examples and solved problems.
2. Discrete Distributions
mean
In this section, we will study some of the discrete probability distributions, their
, variance
and the associated m.g.f.
Recall that, a random variable
is discrete, if
takes distinct values such as
i.e., if the range of
is countable. The corresponding density function
is
known as discrete density function of given by
Discrete Uniform Distribution
Definition 2.1 A random variable
is said to have discrete uniform distribution if the
discrete density function of is given by
The next result gives the mean, variance and m.g.f. of discrete uniform distribution.
Theorem 2.2 If a random variable
has a discrete uniform distribution, then
and
.
Proof. We have
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Discrete and Continuous Distributions
Now, using (1) and (2), we have
The m.g.f. of
is given by
Binomial Distribution and Bernoulli distribution
Definition 2.3 A random variable
density function of
is said to have a Binomial distribution if the discrete
is given by
.
Here,
and
such that
and
are the two parameters of the Binomial
distribution, where represents the probability of success and represents the probability
of failure in a single trial.
We represent Binomial distribution by these notations
or
.
Note that the name “Binomial distribution” comes from the fact that the values of
(or
) for
are nothing but the successive terms of the Binomial expansion of
i.e.,
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Discrete and Continuous Distributions
Also, note that the sum of the probabilities is equal to 1 since
If
, the binomial distribution is known as Bernoulli distribution i.e., a random
variable
is said to have a Bernoulli distribution if the discrete density function of
is
given by
We represent Bernoulli distribution by the notation
Here,
and
.
.
The next result gives the mean, variance and m.g.f. of discrete Binomial distribution.
Theorem 2.4 If a random variable
has a Binomial distribution, then
and
.
Proof. We have
Therefore,
Now,
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Discrete and Continuous Distributions
Now, using (1) and (2), we have
The m.g.f. of
is given by
The next result gives the mean, variance and m.g.f. of Bernoulli distribution.
Theorem 2.5 If a random variable
has a Bernoulli distribution, then
and
.
Proof. We have
This gives
The m.g.f. of
is given by
Let us see some of the examples.
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Discrete and Continuous Distributions
Example 2.6 If Six dice are thrown
three dice to show a or ?
Solution. Let
times, then how many times do you expect at least
denotes the probability of getting
or .
Then,
.
The probability of
successes is given by
Now, the probability of getting at least three successes is
Thus, the number of times at least 3 successes occur
Example 2.7 Let the m.g.f. of a random variable
Then, find
and m.g.f. of
.
Solution. Since
is of the form
, using Theorem 5.4, we have
.
Again, using Theorem 4.3 (i), we have
Poisson Distribution
Definition 2.8 A discrete random variable
discrete density function of is given by
where
is said to have Poisson Distribution if the
is known as the parameter of the distribution
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Discrete and Continuous Distributions
The variable
is called Poisson variate and is denoted as
Note that the sum of the probabilities of a Poisson distribution is unity as
The next result gives the mean, variance and m.g.f. of Poisson distribution.
Theorem 2.9 If a random variable
has a Poisson distribution, then
and
.
Proof. We have
Therefore,
Now,
Now, using (1) and (2), we have
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Discrete and Continuous Distributions
The m.g.f. of
is given by
Thus, for a Poisson distribution, we note that
Value Addition
For Binomial Distribution
, let
constant.
Then,
If we let
while
and
remain fixed, then
Thus, the Poisson Distribution is the limiting case of Binomial Distribution.
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Discrete and Continuous Distributions
Example 2.10 The manufacturer of scarf pins knows that
of his product is defective. If
he sells scarf pins in a box of
and guarantees that not more than
pins will be
defective, what is the approximate probability that a box will fail to meet the guaranteed
quality?
Solution. We have
and
Therefore,
Probability of
.
defective scarf pins in a box of
is
Probability that a box will fail to meet the guaranteed quality is
Geometric Distribution
Definition 2.11 A random variable
density function of is given by
Where
Here,
is said to have geometric distribution if the discrete
and
is known as the parameter of the geometric distribution.
Note that since the various terms in the discrete density function of
are
which clearly form a geometric series, the distribution
is known as geometric
distribution.
The next result gives the mean, variance and m.g.f. of geometric distribution.
Theorem 2.12 If a discrete random variable
has a geometric distribution, then
and
Proof. We have
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Discrete and Continuous Distributions
Therefore,
Now,
Now,
Differentiating w.r.t. , we have
Therefore,
Now, using (1) and (2), we have
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Discrete and Continuous Distributions
The m.g.f. of
is given by
Example 2.13 A die is thrown until a side with a six appears. Then, find the probability that
it must be thrown more than four times?
Solution. The probability of getting a six is
If
, so
.
denotes the number of throws required for the first success, then
for
The required probability is given by
Negative Binomial Distribution
Definition 2.14 A discrete random variable
is said to have negative binomial
distribution if the discrete density function of is given by
,
where
and
are known as the parameters and
denote negative binomial distribution as
.
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We
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Discrete and Continuous Distributions
Value Addition
1. If we take
, the negative binomial distribution becomes a geometric distribution.
2. Consider
Thus, the negative binomial distribution is also given by
The next result gives the mean, variance and m.g.f. of negative binomial distribution
distribution.
Theorem 2.15 If a discrete random variable
distribution, then
has a negative binomial distribution
and
Proof. The m.g.f. of
is given by
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Discrete and Continuous Distributions
Differentiating twice
w.r.t. , we have
Therefore, we have
and
Now, using (1) and (2), we have
Example 2.16 A man participating in an archery tournament is continuously trying to hit a
target. What is the probability that his tenth trial is his fifth hit, if the probability of hitting
the target at any trial is
Solution. Here, we have
If
?
, so
.
denotes the number of failures preceding the
Clearly,
success.
and since
, so
success, then
is the number of failures preceding the
Therefore, we have
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Discrete and Continuous Distributions
Some Solved Problems
Problem 7 An experiment succeeds twice as often as is fails. Find the chance that in the
next six trials there will be at least successes.
Solution. Clearly, the probability of getting success
.
Thus,
The probability of
successes is given by
So, the required probability
Problem 8 Let
value of .
be a binomial variate and if
Solution. Now, the probability of
Then, for
Problem 9 Let
and
, then find the
successes is given by
, we have
be a Poisson variate such that
Then, find
Solution. Using Definition 5.7, we have
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Discrete and Continuous Distributions
Thus,
Problem 10 Suppose that the average number of customer’s phone calls at the customer
care centre of a well-known telecom company regarding complaints about the network issue
related problems is
calls per hour. Use Poisson distribution to answer the following
questions:
(i) What is the probability that no calls will arrive in a -minute interval?
(ii) What is the probability that more than five calls will arrive in a -minute interval?
Solution. (i) Average number of customer’s phone calls in a -minute interval is
Probability that no calls will arrive in a -minute interval is
(ii) Average number of customer’s phone calls in a -minute interval is
Probability that more than five calls will arrive in a -minute interval is
Problem 11 If
not?
and
, then whether
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Discrete and Continuous Distributions
Solution. Here,
and
Now, consider
But in this case,
Hence
which is not possible.
can not have negative binomial distribution.
3. Continuous Distributions
In this section, we will study some of the continuous probability distributions, their
mean
, variance
and the associated m.g.f.
Uniform Distribution
Definition 3.1 A continuous random variable is said to have uniform distribution over
the interval
, where
, if its probability density function p.d.f. is given
by
We denote uniform distribution by
Theorem 3.2 If a continuous random variable
has a uniform distribution over
, then
and
Proof. We have
Now,
Now, using (1) and (2), we have
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Discrete and Continuous Distributions
The m.g.f. of
is given by
Example 3.3 If a random variable
for which
Also compute
Solution. The p.d.f.
has a uniform distribution over
. Then, find
.
is given by
Now,
We are given that
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Discrete and Continuous Distributions
Now,
Exponential Distribution
Definition 3.4 A continuous random variable
is said to have an exponential
distribution with parameter
, if its p.d.f.
is given by
, for all
and
We denote exponential distribution by
Theorem 3.5 If a continuous random variable
and
has exponential distribution, then
, for
Proof. We have
Now,
Now, using (1) and (2), we have
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Discrete and Continuous Distributions
The m.g.f. of
is given by
Example 3.6 If
with
, then find
.
Solution. We have
Using Theorem 3.5, we have
Example 3.7 If
Then, find the value of
and
such that
.
Solution. We have
Since
, so we have
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Discrete and Continuous Distributions
Thus,
Normal Distribution
Definition 3.8 A continuous random variable
probability density function is given by
is said to have normal distribution if its
Random variable
is known as normal variate and
distribution . We denote normal distribution by
The cumulative distribution function (c.d.f.) of
and
are the parameters of normal
.
is denoted as
and is given by
Value Addition
1. The curve
is known as the normal curve. It is a bell-shaped curve
which is symmetrical about the line
If we increases
numerically, then
value of
which occurs at
is
decreases rapidly and we note that the maximum
One may note that since the ordinate at
divides the area under the normal curve into two equal parts, the median of the curve is
at the
and in fact, mean, median and mode of the normal distribution coincides.
2. The total area under the curve is
since
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Discrete and Continuous Distributions
where
is a gamma function and
Theorem 3.9 If a continuous random variable
has normal distribution, then
and
Proof. The M.G.F. of
Thus, the M.G.F. of
is given by
is
Differentiating (1) w.r.t. , we have
Therefore,
Differentiating (2) w.r.t. , we have
Thus,
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Discrete and Continuous Distributions
Now, using (3) and (4), we have
Definition 3.10 If
denotes the normal distribution i.e.,
, then
is known
as the standard normal variate with
and
and is denoted as
The probability density function of the standard normal variate is given by
Note that since
, we have
Therefore,
.
The corresponding cumulative distribution function is given by
Theorem 3.11 Let
denotes the normal distribution i.e.,
, then
Proof. We have
Taking
If
put
and if
put
This gives
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Discrete and Continuous Distributions
The cumulative distribution function of
is given by
Now, consider
Using (1) and (2), we have
Corollary 3.12 Let
denotes the standard normal distribution i.e.,
, then
Proof. The proof follows from the Theorem 3.10.
Value Addition
Let
be a standard normal variate with probability density function
The area under the standard normal curve between the ordinates
by the definite integral
given by
to
is given
The above integral is known as the normal probability integral.
Note that
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Discrete and Continuous Distributions
Therefore,
This gives
Thus,
(by symmetry)
Example 3.13 Let
Then,
be a normal variate with mean
and standard deviation
.
(using Normal Table)
Example 3.14 The life of LED bulbs of a certain brand may be assumed to be normally
distributed with mean
days and standard deviation
days. What is the probability
(i) that the life of a randomly chosen LED bulb is less than
days.
(ii) that the total life of a randomly chosen LED bulb is between 136 days and 174 days.
Solution. (i) We have
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Discrete and Continuous Distributions
(by symmetry)
(using Normal Table)
(ii) We have
(using Normal Table)
Exercises
1.
2.
3.
4.
What is the probability of obtaining
heads in a toss of
coins and then
repeating this experiment two times in the next five tosses ?
A family has
children. Find the probability that this family has at least one girl,
given that they have at least one boy. Assume that either sex-birth is to equally
likely to occur and all births were independent.
Suppose
balls are distributed at random into
boxes. Find the probability that
there are exactly balls in the first
boxes.
Assume that you have a fair die. How many times should you roll it so that with a
probability of at least
5.
the frequency of six’s will differ from by less than
The number of criminals being hanged in a week is a Poisson variate with mean .
However,
itself is a variate which takes on one of the values
with
respective probabilities
6.
7.
Suppose that the number of trucks passing an intersection obeys Poisson
distribution. If the probability of no truck in one minute is
, what is the
probability of more than one car in two minutes ?
and
shoot independently until each has hit his own target. They have
probabilities
8.
9.
What is the probability that no criminal is hanged.
and
of hitting the targets at each shot respectively. Find the
probability that will require more shots than
Find the probability that a person tossing coins will get either all heads or all tails
for the second time on the sixth toss.
A man and a woman agree to meet at a certain place between
PM and
PM.
They agree that one arriving first will wait
hours,
, for other to arrive.
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Discrete and Continuous Distributions
Assuming that the arrival times are independent and uniformly distributed, find the
probability that they meet.
10. The daily consumption of milk in excess of
gallons is approximately
exponentially distributed with
. The city has a daily stock of
gallons.
What is the probability that of two days selected at random, the stock is insufficient
for both the days ?
11. In a large group of men,
are under
inches in height and
are between
inches. Assuming a normal distribution, find the mean height and standard
deviation.
12. If is
, for what value of ,
is minimized.
References
1. Robert V. Hogg, Joseph W. McKean and Allen T. Craig, Introduction to Mathematical
Statistics, Pearson Education, Asia, 2007.
2. Irwin Miller and Marylees Miller, John E. Freund’s Mathematical Statistics
with Applications (7th Edition), Pearson Education, Asia, 2006.
3. Sheldon Ross, Introduction to Probability Models (9th Edition), Academic Press,
Indian Reprint, 2007.
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