PLEASE ANSWER ALL QUESTIONS

3.105. Let X be a random variable that can take on the values 2, 1, and 3 with respective probabilities 1 3, 1 6, and 1 2. Find (a) the mean, (b) the variance, (c) the moment generating function, (d) the characteristic function, (e) the third moment about the mean.3.105. Let X be a random variable that can take on the values 2, 1, and 3 with respective probabilities 1 3, 1 6, and 1 2. Find (a) the mean, (b) the variance, (c) the moment generating function, (d) the characteristic function, (e) the third moment about the mean.

3.108. Let X be a random variable having density function where c is an appropriate constant. Find (a) the mean, (b) the variance, (c) the moment generating function, (d) the characteristic function, (e) the coefficient of skewness, (f) the coefficient of kurtosis. 3.109. Let X and Y have joint density function Find (a) E(X2 Y2), (b) 3.110. Work Problem 3.109 if X and Y are independent identically distributed random variables having density function f(u) (2p)1>2eu2>2

Suppose that we have an experiment such as tossing a coin or die repeatedly or choosing a marble from an urn repeatedly. Each toss or selection is called a trial. In any single trial there will be a probability associated with a particular event such as head on the coin, 4 on the die, or selection of a red marble. In some cases this probability will not change from one trial to the next (as in tossing a coin or die). Such trials are then said to be independent and are often called Bernoulli trials after James Bernoulli who investigated them at the end of the seventeenth century

Let p be the probability that an event will happen in any single Bernoulli trial (called the probability of success). Then q 1 p is the probability that the event will fail to happen in any single trial (called the probability of failure). The probability that the event will happen exactly x times in n trials (i.e., successes and n x failures will occur) is given by the probability function

In other words, in the long run it becomes extremely likely that the proportion of successes, , will be as close as you like to the probability of success in a single trial, p. This law in a sense justifies use of the empirical definition of probability on page 5. A stronger result is provided by the strong law of large numbers (page 83), which states that with probability one, , i.e., actually converges to p except in a negligible number of cases.

If n is large and if neither p nor q is too close to zero, the binomial distribution can be closely approximated by a normal distribution with standardized random variable given by (11) Here X is the random variable giving the number of successes in n Bernoulli trials and p is the probability of success. The approximation becomes better with increasing n and is exact in the limiting case. In practice, the approximation is very good if both np and nq are greater than 5

In the binomial distribution (1), if n is large while the probability p of occurrence of an event is close to zero, so that q 1 p is close to 1, the event is called a rare event. In practice we shall consider an event as rare if the number of trials is at least 50 (n 50) while np is less than 5. For such cases the binomial distribution is very closely approximated by the Poisson distribution (13) with np. This is to be expected on comparing Tables 4-1 and 4-3, since by placing np, q 1, and p 0

e independent random variables that are identically distributed (i.e., all have the same probability function in the discrete case or density function in the continuous case) and have finite mean and variance 2