Download Mar 23 (Lecture 6)

Ch5. Probability Densities
Dr. Deshi Ye
[email protected]
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Outline
 Continuous Random variables
 Kinds of Probability distribution
 Normal distr. Uniform distr. Log-Normal
dist. Gamma distr. Beta distr. Weibull
distr.
 Joint distribution
 Checking data if it is normal?
 Transform observation to near normal
 Simulation
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5.1 Continuous Random Variables
 Continuous sample space: the speed
of car, the amount of alcohol in a
person’s blood
 Consider the probability that if an
accident occurs on a freeway whose
length is 200 miles.
 Question: how to assign probabilities?
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Assign Prob.
 Suppose we are interested in the prob.
that a given random variable will take on
a value on the interval [a, b]
 We divide [a, b] into n equal
subintervals of width ∆x,Frequency
b – a = n ∆x,
containing the points x1, x2, ..., xn,
respectively.
n
 Then
P(a  x  b)   f ( xi )  x
i 1
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 If f is an integrable function for all
values of the random variable, letting
∆x-> 0, then
b
P(a  x  b)   f ( x)dx
a
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Continuous Probability Density
Function
 1. Shows All Values, x,
& Frequencies, f(x)
Frequency
 f(X) Is Not Probability
 2. Properties
(Value, Frequency)
f(x)
 f (x )dx  1
All X
(Area Under Curve)
f ( x )  0, a  x  b
a
x
b
Value
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Continuous Random Variable
Probability
b
Probability Is Area
Under Curve!
P(a  x  b)   f ( x)dx
a
f(x)
a
b
X
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Distribution function F
 Distribution function F (cumulative
distribution )
x
F ( x)   f (t )dt

Or
P( X  x)
Integral calculus:
dF ( x)
 f ( x)
x
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EX
 If a random variable has the
probability density
2e 2 x for x  0
f ( x)  
else
0
find the probabilities that it will take
on a value
A) between 1 and 3 B) greater than
0.5
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Solution
A)
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P(1  x  3)   2e2 x dx  e2 x |13  e6  e2  0.133
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B)

P( x  0.5)   2e2 x dx  e2 x |0.5  0  e1  0.368
0.5
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Mean and Variance
Mean:

   xf ( x)dx

Variance:

   ( x   ) f ( x)dx
2
2

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K-th moment
 About the original

   x  f ( x) dx
'
k
k

 About the mean

k   ( x   )  f ( x) dx
k

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Useful cheat
n ax
xe
n n1 ax
x
e
dx


x
e
dx


a
a
n ax
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Continuous Probability Distribution
Models
Continuous
Probability Distribution
Uniform
Normal
Exponential
Others
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Normal Distribution
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5.2 The Normal Distribution
 Normal probability density (normal
distribution)

1
f ( x;  ,  2 ) 
e
2 
( x )2
2 2
   x  
The mean and variance of normal distribution is exactly
 and 
2
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The Normal Distribution
 1. ‘Bell-Shaped’ &
Symmetrical
f(X)
 2. Mean, median,
mode are equal
 3. Random variable
has infinite range
X
Mean
Median
Mode
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The Normal Distribution

1
f ( x;  ,  ) 
e
2 
( x )2
2
f(x)
 =
 =
x =
)
 =
2 2
   x  
= Frequency of random variable x
Population standard deviation
3.14159; e = 2.71828
value of random variable (- < x <
Population mean
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Effect of varying parameters
( & )
f(X)
B
A
C
X
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Standard normal distribution
function
 Standard normal distribution, with
mean 0 and variance 1. Hence
1
P( Z  z )  F ( z ) 
2
z
e

P(a  x  b)  F (b)  F (a)
t 2 / 2
dt
Normal
table
F ( z )  1  F ( z )
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Standardize the
Normal Distribution
X 
Z

Normal
Distribution
Standardized
Normal Distribution
= 1


X
=0
Z
One table!
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Not standard normal
distribution
 Let
Z
X u
, then the random

Variable Z, F(z) has a standard normal
distribution. We call it z-scores.
 When X has normal distribution with
mean  and standard deviation 
P ( a  x  b)  F (
b

)  F(
a

)
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Find z values for the known
probability
 Given probability relating to standard
normal distribution, find the
corresponding value z.
 F(z) is known, what is the value of z?
 Let z be such that probability is 
where
  P(Z  z )
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Finding Z Values
for Known Probabilities
Standardized Normal
Probability Table (Portion)
What is Z given
P(Z) = .1217?
.1217
=1
Z
.00
.01
0.2
0.0 .0000 .0040 .0080
0.1 .0398 .0438 .0478
 = 0 .31
Shaded area
exaggerated
Z
0.2 .0793 .0832 .0871
0.3
.1179 .1217 .1255
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F (z )  1  
Find the following values (check it in Table)
F ( z0.01 )  1  0.01  0.99, z0.01  2.33
F ( z0.05 )  1  0.05  0.95, z0.05  1.645
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5.3 The Normal Approximation to
the binomial distribution
 Theorem 5.1. If X is a random variable
having the binomial distribution with
parameter n and p, the limiting form of
the distribution function of the
X  np
standardized random variable Z 
np(1  p)
 as n approaches infinity, is given by the
standard normal distribution
F ( z)  
z

1 t 2 / 2
e
dt    z  
2
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EX
 If 20% of the memory chips made in
a certain plant are defective, what are
the probabilities that in a lot of 100
random chosen for inspection?
 A) at most 15.5 will be defective
 B) exactly 15 will be defective
 Hint: calculate it in binomial dist. And
normal distribution.
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A good rule
 A good rule for normal approximation
to the binomial distribution is that
both
np and n(1-p) is at least 15
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