Download Normal Distribution

Continuous Probability
Distributions
Fangrong Yan
Outline
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Introduction
General Concepts
The Normal Distribution
Properties of the Standard Normal Distribution
Conversion from an standard Normal Distribution
Linear Combinations of Random Variables
Normal Approximation to the Binomial Distribution
Normal Approximation to the Poisson Distribution
Introduction
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This chapter discusses continuous probability
distributions. Specifically, the normal
Distribution the most widely used distribution
in statistical work—is explored in depth.
Continuous Random Variable
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A continuous random variable is one for
which the outcome can be any value in an
interval of the real number line.
Usually a measurement.
Examples
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Let Y = length in mm
Let Y = time in seconds
Let Y = temperature in ºC
Continuous Random Variable
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We don’t calculate P(Y = y), we calculate P(a
< Y < b), where a and b are real numbers.
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For a continuous random variable
P(Y = y) = 0.
Continuous Random Variables
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The probability density function (pdf) when
plotted against the possible values of Y forms a
curve. The area under an interval of the curve is
equal to the probability that Y is in that interval.
0.40
f(y)
a
b Y
The entire area under a probability density curve for a
continuous random variable
A.
B.
C.
D.
Is always greater than 1.
Is always less than 1.
Is always equal to 1.
Is undeterminable.
Properties of a Probability Density Function (pdf)
1)
f(y) > 0 for all possible intervals of y.

2)
 f ( y )dy  1

3)
If y0 is a specific value of interest, then the
cumulative distribution function (cdf) is
F ( y0 )  P (Y  y0 ) 
y0
 f ( y)dy

4)
If y1 and y2 are specific values of interest, then
P( y1  Y  y2 ) 
y2
 f ( y)dy  F ( y )  F ( y )
2
y1
1
Suppose a random variable Y has the following
probability density function:
f(y) = y if 0<y<1
2-y if 1 < y<2
0 if 2 < y.
Find the complete form of the cumulative
distribution function F(y) for any real value y.
Random Variable and Distribution
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A random variable X is a numerical outcome of a
random experiment
The distribution of a random variable is the collection
of possible outcomes along with their probabilities:
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Pr( X  x)  p ( x)
Discrete case:
b
Continuous case: Pr(a  X  b)  a p ( x)dx
Expected Value for a Continuous
Random Variable
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Recall Expected Value for a discrete random variable:
E(Y )     y  p( y)
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Expected value for a continuous random variable:
E (Y )   

yf
(
y
)
dy


Variance for Continuous Random
Variable
Recall: Variance for a discrete random
variable:
Var(Y )     ( y   ) p( y)
2
2
Variance for a continuous random variable:

Var (Y )     ( y   ) f ( y )dy
2
2

Mean and Variance of a Continuous
Random Variable
Definition
Continuous Uniform Random Variable
Figure 4-1 Continuous uniform probability density
function.
Continuous Uniform Random Variable
Mean and Variance
Continuous Uniform Random Variable
Example
Continuous Uniform Random Variable
Figure 4-2Probability for Example 4-9.
Continuous Uniform Random Variable
Normal Distribution
Definition
Normal Distribution
Figure 4-10 Normal probability density functions for
selected values of the parameters  and 2.
Normal Distribution
Some useful results concerning the normal distribution
Normal Probability Distribution
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Characteristics
68.26% of values of a normal random variable
are within +/- 1 standard deviation of its mean.
95.44% of values of a normal random variable
are within +/- 2 standard deviations of its mean.
99.72% of values of a normal random variable
are within +/- 3 standard deviations of its mean.
Normal Probability Distribution
99.72%
95.44%
68.26%
 – 3
 – 1
 – 2

 + 3
 + 1
 + 2
x
Normal Distribution
Definition : Standard Normal
Normal Distribution
Example 4-11
Figure 4-13 Standard normal probability density function.
Normal Distribution
Standardizing
Normal Distribution
Example 4-13
Looking up probabilities in the standard
normal table
What is the area to the
left of Z=1.51 in a
standard normal curve?
Z=1.51
Z=1.51
Area is 93.45%
Normal Distribution
Figure 4-3 Standardizing a normal random variable.
Exercise 1
a) What is P(z ≤2.46)?
b) What is P(z ≥2.46)?
Answer:
a) .9931
b) 1-.9931=.0069
2.46
z
Exercise
2
a) What is P(z ≤-1.29)?
Answer:
b) What is P(z ≥-1.29)?
a) 1-.9015=.0985
b) .9015
Red-shaded area is
equal to greenshaded area
Note that:
P ( z  1.29)  1  P ( z  1.29)
-1.29
1.29
z
Note that, because of the symmetry, the area to the left of -1.29 is
the same as the area to the right of 1.29
What is P(.00 ≤ z ≤1.00)?
Exercise 3
P(.00 ≤ z ≤1.00)=.3413
0
1
z
P(.00  z  1)  P( z  1)  P( z  0)
 .8413  .5000  .3413
What is P(-1.67 ≥ z ≥ 1.00)?
Exercise 4
P(-1.67 ≤ z ≤1.00)=.7938
Thus P(-1.67 ≥ z ≥ 1.00)
=1 - .7938 = .2062
-1.67
0
1
z
P ( 1.67  z  1)  P ( z  1)  [1  P ( z  1.67)]
 .8413  (1  .9525)  .7938
Normal Distribution
To Calculate Probability
Normal Distribution
Example 4-14
Normal Distribution
Example 4-14 (continued)
Normal Distribution
Example 4-14 (continued)
Figure 4-16 Determining the value of x to meet a
specified probability.
Normal Approximation to the Binomial
and Poisson Distributions
• Under certain conditions, the normal
distribution can be used to approximate the
binomial distribution and the Poisson
distribution.
Normal Approximation to the Binomial
and Poisson Distributions
Figure 4-19 Normal
approximation to the
binomial.
Normal Approximation to the Binomial
and Poisson Distributions
Example 4-17
Normal Approximation to the Binomial
and Poisson Distributions
Normal Approximation to the Binomial Distribution
Normal Approximation to the Binomial
and Poisson Distributions
Example 4-18
Normal Approximation to the Binomial
and Poisson Distributions
Figure 4-21 Conditions for approximating hypergeometric
and binomial probabilities.
Normal Approximation to the Binomial
and Poisson Distributions
Normal Approximation to the Poisson Distribution
Normal Approximation to the Binomial
and Poisson Distributions
Example 4-20
Exponential Distribution
Definition
Exponential Distribution
Mean and Variance
Exponential Distribution
Example 4-21
Exponential Distribution
Figure 4-23 Probability for the exponential distribution in
Example 4-21.
Exponential Distribution
Example 4-21 (continued)
Exponential Distribution
Example 4-21 (continued)
Expectation
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A random variable X~Pr(X=x). Then, its expectation is
E[ X ]   x x Pr( X  x)
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In an empirical sample, x1, x2,…, xN,
1
N
E[ X ]  i 1 xi
N
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Continuous case:
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E[ X ]   xp ( x)dx

Expectation of sum of random variables
E[ X1  X 2 ]  E[ X1 ]  E[ X 2 ]
Expectation: Example
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Let S be the set of all sequence of three rolls of a die.
Let X be the sum of the number of dots on the three
rolls.
What is E(X)?
Let S be the set of all sequence of three rolls of a die.
Let X be the product of the number of dots on the
three rolls.
What is E(X)?