Today Today: More Chapter 5 Reading: –Important Sections in Chapter 5: 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 Suppose X and Y have joint pdf f(x,y)=x+y for 0<x<1, 0<y<1 Find the marginal distributions of X and Y Is there a linear relationship between X and Y?

Covariance and Correlation Recall, the covariance between tow random variables is: The covariance is:

Properties Cov(X,Y)=E(XY)-μ X μ Y Cov(X,Y)=Cov(Y,X) Cov(aX,bY)=abCov(X,Y) Cov(X+Y,Z)=Cov(X,Z)+Cov(Y,Z)

Example Suppose X and Y have joint pdf f(x,y)=x+y for 0<x<1, 0<y<1 Is there a linear relationship between X and Y? What is Cov(3X,-4Y)? What is the correlation between 3X and -4Y

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:

Example Suppose X and Y have joint pdf f(x,y)=x+y for 0<x<1, 0<y<1 Are X and Y independent?

Example Suppose X and Y have joint pdf f(x,y)=x+y for 0<x<1, 0<y<1 Are X and Y independent?

Hard Example Suppose X and Y have joint pdf f(x,y)=45x 2 y 2 for |x|+|y|<1 Are X and Y independent? What is their covariance?

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:

Example Suppose X and Y have joint pdf f(x,y)=x+y for 0<x<1, 0<y<1 What is the conditional distribution of X given Y=y? Find the probability that X<1/2 given Y=1/2

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:

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

CDF of the Normal Distribution The cdf of a continuous random variable,Z, is F(z)=P(Z<=z) For the standard normal distribution this is

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 σ has a normal distribution only if the z-score has a standard normal

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

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 99 th percentile of this distribution?

Finding a Percentile Can use the relationship between Z and the random variable X to compute percentiles for the distribution of X The 100p th percentile of normally distributed random variable X with mean μ and variance σ can be found using the standard normal distribution

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 99 th percentile of this distribution?