Introduction to Probability Modeling – Chapter 02

Random Variables

Example 2.8 Suppose that an airplane engine will fail, when in flight, with probability 1−p independently from engine to engine; suppose that the airplane will make a successful flight if at least 50 percent of its engines remain operative. For what values of p is a four-engine plane preferable to a two-engine plane?

Example 2.9 Suppose that a particular trait of a person (such as eye color or left handedness) is classified on the basis of one pair of genes and suppose that d represents a dominant gene and r a recessive gene. Thus a person with dd genes is pure dominance, one with rr is pure recessive, and one with rd is hybrid. The pure dominance and the hybrid are alike in appearance. Children receive one gene from each parent. If, with respect to a particular trait, two hybrid parents have a total of four children, what is the probability that exactly three of the four children have the outward appearance of the dominant gene?

Example 2.10 Suppose that the number of typographical errors on a single page of this book has a Poisson distribution with parameter λ = 1. Calculate the probability that there is at least one error on this page.

Example 2.12 (α-particles ) Consider an experiment that consists of counting the number of α-particles given off in a one-second interval by one gram of radioactive material. If we know from past experience that, on the average, 3.2 such α-particles are given off, what is a good approximation to the probability that no more than two α-particles will appear?

Example 2.18 (Expectation of a Geometric Random Variable) Calculate the expectation of a geometric random variable having parameter p.

Example 2.31 At a party N men throw their hats into the center of a room. The hats are mixed up and each man randomly selects one. Find the expected number of men who select their own hats.

Example 2.32 Suppose there are 25 different types of coupons and suppose that each time one obtains a coupon, it is equally likely to be any one of the 25 types. Compute the expected number of different types that are contained in a set of 10 coupons.

Example 2.37 (Sums of Independent Poisson Random Variables) Let X and Y be independent Poisson random variables with respective means λ1 and λ2. Calculate the distribution of X + Y.

Example 2.38 (Order Statistics) Let X(1), . . . , X(n) be independent and identically distributed continuous random variables with probability distribution F and density function F’ = f . If we let X(i) denote the ith smallest of these random variables, then X(1), . . . , X(n)are called the order statistics. To obtain the distribution of X(i), note that X(i) will be less than or equal to x if and only if at least i of the n random variables X(1), . . . , X(n) are less than or equal to x.

Example 2.49 Suppose we know that the number of items produced in a factory during a week is a random variable with mean 500 1.What can be said about the probability that this week’s production will be at least 1000? (Answer = 0.5).
2.If the variance of a week’s production is known to equal 83333.3, then what can be said about the probability that this week’s production    will be between 400 and 600? (Answer = 1).

Example 2.52 The lifetime of a special type of battery is a random variable with mean 40 hours and standard deviation 20. hours  A battery is used until it fails, at which point it is replaced by a new  ne. Assuming a stockpile of 25 such batteries, the lifetimes of which are independent, approximate the probability that over 1100 hours of use can be obtained.

Example 2.53 Consider a particle that moves along a set of m + 1 nodes, labeled 0, 1, . . . ,m, that are arranged around a circle. At each step the particle is equally likely to move one position in either the clockwise or counterclockwise direction. That is, if Xn is the position of the particle after its nth step then
P{Xn+1 = i + 1|Xn = i} = P{Xn+1 = i − 1|Xn = i} = 1/2 where i + 1 ≡ 0
When i = m, and i − 1 ≡ m when i = 0. Suppose now that the particle starts at 0 and continues to move around according to the preceding rules until all the nodes 1, 2, . . . ,m have been visited. What is the probability that node i, i = 1, . . . ,m, is the last one visited?

Exercise 2.02 Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed n times. What are the possible values of X?

Exercise 2.28 – Making a biased system useful! Suppose that we want to generate a random variable X that is equally likely to be either 0 or 1, and that all we have at our disposal is a biased coin that, when flipped, lands on heads with some (unknown) probability p. Consider the following procedure:
1.Flip the coin, and let 01, either heads or tails, be the result
2.Flip the coin again, and let 02 be the result
3.If 01 and 02 are the same, return to step 1
4.If 02 is heads, set X = 0, otherwise set X = 1