Transcript Lecture 13 - Statistics

Today
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Today: More Chapter 5
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Reading:
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Important Sections in Chapter 5: 5.1-5.11
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Only material covered in class
Note we have not, and will not cover moment/probability generating
functions
Suggested problems: 5.1, 5.2, 5.3, 5.15, 5.25, 5.33, 5.38, 5.47, 5.53, 5.62
Exam will be returned in Discussion Session
Important Sections in Chapter 5
Example
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Suppose X and Y have joint pdf f(x,y)=x+y for 0<x<1, 0<y<1
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Find the marginal distributions of X and Y
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Is there a linear relationship between X and Y?
Covariance and Correlation
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Recall, the covariance between tow random variables is:
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The covariance is:
Properties
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Cov(X,Y)=E(XY)-μXμY
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Cov(X,Y)=Cov(Y,X)
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Cov(aX,bY)=abCov(X,Y)
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Cov(X+Y,Z)=Cov(X,Z)+Cov(Y,Z)
Example
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Suppose X and Y have joint pdf f(x,y)=x+y for 0<x<1, 0<y<1
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Is there a linear relationship between X and Y?
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What is Cov(3X,-4Y)?
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What is the correlation between 3X and -4Y
Independence
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In the discrete case, two random variables are independent if:
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In the continuous case, X and Y are independent if:
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If two random variables are independent, their correlation
(covariance) is:
Example
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Suppose X and Y have joint pdf f(x,y)=x+y for 0<x<1, 0<y<1
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Are X and Y independent?
Example
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Suppose X and Y have joint pdf f(x,y)=x+y for 0<x<1, 0<y<1
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Are X and Y independent?
Hard Example
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Suppose X and Y have joint pdf f(x,y)=45x2y2 for |x|+|y|<1
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Are X and Y independent?
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What is their covariance?
Conditional Distributions
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Similar to the discrete case, we can update our probability function if
one of the random variables has been observed
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In the discrete case, the conditional probability function is:
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In the continuous case, the conditional pdf is:
Example
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Suppose X and Y have joint pdf f(x,y)=x+y for 0<x<1, 0<y<1
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What is the conditional distribution of X given Y=y?
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Find the probability that X<1/2 given Y=1/2
Normal Distribution
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One of the most important distributions is the Normal distribution
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This is the famous bell shaped distribution
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The pdf of the normal distribution is:
f ( x) 
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1
2 
1
( x )2
2
2

e
Where the mean and variance are:
Normal Distribution
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A common reference distribution (as we shall see later in the course)
is the standard normal distribution, which has mean 0 and variance of
1
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The pdf of the standard normal is:
f ( z) 
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1
2 
z2
2
e 2
Note, we denote the standard normal random variable by Z
CDF of the Normal Distribution
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The cdf of a continuous random variable,Z, is F(z)=P(Z<=z)
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For the standard normal distribution this is
z
( z ) 


1
2 
u 2
2
e 2 du
Relating the Standard Normal to Other Normal Distributions
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Can use the standard normal distribution to help compute
probabilities from other normal distributions
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This can be done using a z-score:
Z
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(X  )

A random variable X with mean μ and variance σ has a normal
distribution only if the z-score has a standard normal
Relating the Standard Normal to Other Normal Distributions
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If the z-score has a standard normal distribution, can use the standard
normal to compute probabilities
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Table II gives values for the cdf of the standard normal
Example:
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The height of female students at a University follows a normal
distribution with mean of 65 inches and standard deviation of 2
inches
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Find the probability that a randomly selected female student has a
height less than 58 inches
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What is the 99th percentile of this distribution?
Finding a Percentile
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Can use the relationship between Z and the random variable X to
compute percentiles for the distribution of X
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The 100pth percentile of normally distributed random variable X with
mean μ and variance σ can be found using the standard normal
distribution
Example:
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The height of female students at a University follows a normal
distribution with mean of 65 inches and standard deviation of 2
inches
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What is the 99th percentile of this distribution?