Today

1
Today

• Today More Chapter 5
• Reading
• Important Sections in Chapter 5 5.1-5.11
• 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

2
Example

• Suppose X and Y have joint pdf f(x,y)xy for
  0ltxlt1, 0ltylt1
• Find the marginal distributions of X and Y
• Is there a linear relationship between X and Y?

3
Covariance and Correlation

• Recall, the covariance between tow random
  variables is
• The covariance is

4
Properties

• Cov(X,Y)E(XY)-µXµY
• Cov(X,Y)Cov(Y,X)
• Cov(aX,bY)abCov(X,Y)
• Cov(XY,Z)Cov(X,Z)Cov(Y,Z)

5
Example

• Suppose X and Y have joint pdf f(x,y)xy for
  0ltxlt1, 0ltylt1
• Is there a linear relationship between X and Y?
• What is Cov(3X,-4Y)?
• What is the correlation between 3X and -4Y

6
Independence

• In the discrete case, two random variables are
  independent if
• In the continuous case, X and Y are independent
  if
• If two random variables are independent, their
  correlation (covariance) is

7
Example

• Suppose X and Y have joint pdf f(x,y)xy for
  0ltxlt1, 0ltylt1
• Are X and Y independent?

8
Example

• Suppose X and Y have joint pdf f(x,y)xy for
  0ltxlt1, 0ltylt1
• Are X and Y independent?

9
Hard Example

• Suppose X and Y have joint pdf f(x,y)45x2y2 for
  xylt1
• Are X and Y independent?
• What is their covariance?

10
Conditional Distributions

• Similar to the discrete case, we can update our
  probability function if one of the random
  variables has been observed
• In the discrete case, the conditional probability
  function is
• In the continuous case, the conditional pdf is

11
Example

• Suppose X and Y have joint pdf f(x,y)xy for
  0ltxlt1, 0ltylt1
• What is the conditional distribution of X given
  Yy?
• Find the probability that Xlt1/2 given Y1/2

12
Normal Distribution

• One of the most important distributions is the
  Normal distribution
• This is the famous bell shaped distribution
• The pdf of the normal distribution is
• Where the mean and variance are

13
Normal Distribution

• 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
• The pdf of the standard normal is
• Note, we denote the standard normal random
  variable by Z

14
CDF of the Normal Distribution

• The cdf of a continuous random variable,Z, is
  F(z)P(Zltz)
• For the standard normal distribution this is

15
Relating the Standard Normal to Other Normal
Distributions

• Can use the standard normal distribution to help
  compute probabilities from other normal
  distributions
• This can be done using a z-score
• A random variable X with mean µ and variance s
  has a normal distribution only if the z-score has
  a standard normal

16
Relating the Standard Normal to Other Normal
Distributions

• If the z-score has a standard normal
  distribution, can use the standard normal to
  compute probabilities
• Table II gives values for the cdf of the standard
  normal

17
Example

• The height of female students at a University
  follows a normal distribution with mean of 65
  inches and standard deviation of 2 inches
• Find the probability that a randomly selected
  female student has a height less than 58 inches
• What is the 99th percentile of this distribution?

18
Finding a Percentile

• Can use the relationship between Z and the random
  variable X to compute percentiles for the
  distribution of X
• The 100pth percentile of normally distributed
  random variable X with mean µ and variance s can
  be found using the standard normal distribution

19
Example

• The height of female students at a University
  follows a normal distribution with mean of 65
  inches and standard deviation of 2 inches
• What is the 99th percentile of this distribution?