Conditional Expectations The Law of Large Numbers Special Discrete Distibutions

Conditional ExpectationsSpecial Discrete DistibutionsChapter 4-5DeGroot & Schervish

Conditional Expectation/MeanLet X and Y be random variables such that the mean of Y exists and is finite. The conditional expectation (or conditional mean) of Y given X = x is denoted by E(Y|x) and is defined to be the expectation of the conditional distribution of Y given X = x.Conditional Expectation/Meanif Y has a continuous conditional distribution given X=x with conditional p.d.f. g2(y|x), then

Similarly, if Y has a discrete conditional distribution given X = x with conditional p.f. g2(y|x), then

Example The conditional p.f. of Y given X = 4 is g2(y|4) = f (4, y)/f1(4), which is the x = 4 column of table divided by f1(4) =0.208g2(0|4) = 0.0385, g2(1|4) = 0.5769, g2(2|4) = 0.2885, g2(3|4) = 0.0962.The conditional mean of Y given X = 4 is thenE(Y|4) = 0 0.0385 + 1 0.5769 + 2 0.2885 + 3 0.0962 = 1.442.Theorem Let X and Y be random variables such that Y has finite mean.E[E(Y|X)]= E(Y).Proof

Example Suppose that a point X is chosen on the interval [0, 1]. Also, suppose that after the value X = x has been observed (0 0. A random variable X has the Poisson distribution with mean if the p.f. of X is as follows:

Example Suppose that the average number of accidents occurring weekly on a highway equals 3. Calculate the probability that there is at least one accident this week.Let X denote the number of accidents occurring on the highway in question during this week. Because it is reasonable to suppose that there are a large number of cars passing along that highway, each having a small probability of being involved in an accident, the number of such accidents should be approximately Poisson distributed. Hence,

The Poisson Approximation to Binomial Distributionswhen the value of n is large and the value of p is close to 0, the binomial distribution with parameters n and p can be approximated by the Poisson distribution with mean np.Suppose that in a large population the proportion of people who have a certain disease is 0.01. Determine the probability that in a random group of 200 people at least four people will have the disease.In this example, we can assume that the exact distribution of the number of people having the disease among the 200 people in the random group is the binomial distribution with parameters n = 200 and p = 0.01. Therefore, this distribution can be approximated by the Poisson distribution for which the mean is = np = 2. If X denotes a random variable having this Poisson distribution, then it can be found from the table of the Poisson distribution at the end textbook that Pr(X 4) = 0.1428.Hence, the probability that at least four people will have the disease is approximately 0.1428. The actual value is 0.1420.The Negative Binomial DistributionsSuppose that a machine produces parts that can be either good or defective. Let Xi = 1 if the ith part is defective and Xi = 0 otherwise. Assume that the parts are good or defective independently of each other with Pr(Xi = 1) = p for all i. An inspector observes the parts produced by this machine until she sees four defectives. Let X be the number of good parts observed by the time that the fourth defective is observed. The distribution of X is negative binomial distibution.The Negative Binomial DistributionsSuppose that an infinite sequence of Bernoulli trials with probability of success p are available. The number X of failures that occur before the rth success has the following p.d.f.:

The Geometric DistributionsThe most common special case of a negative binomial random variable is one for which r = 1. This would be the number of failures until the first success.A random variable X has the geometric distribution with parameter p (0