Homework 8 Solutions
1. Ross 6.27 p. 316
Let and be independent random variables. Then the joint density of X and Y is
a)If , the cdf of Z is
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Taking the derivative with respect to a gives us
b)If , the cdf of Z is
Taking the derivative of this with respect to a gives us
.
2. Ross 6.29 p. 316
If , then the cdf of W is
Take the derivative of this with respect to w to obtain the pdf .
3. Ross 6.34 p. 317
a)Let X and Y denote, respectively, the number of males and females in the sample that never eat breakfast. Then X has a binomial distribution with mean and variance , and Y has a binomial distribution with mean and variance . Using the normal approximation to the binomial distribution, and the fact that the sum of 2 normal random variables is itself a normal random variable, we have that is approximately normally distributed with mean 97.6 and variance 73.76; hence, the probability that at least 110 of the 400 people in the sample never eat breakfast is (with the continuity correction)
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b)Let be the difference between the number of females who never eat breakfast and the number of males who never eat breakfast in our sample. Similar to (a), we have that D is approximately normally distributed with mean -3.2 and variance 73.76; hence, the probability that the number of women who never eat breakfast is at least as large as the number of men who never eat breakfast is (again with the continuity correction)
.
4. Ross 6.39 p. 317
a)If X is chosen at random from the set {1, 2, 3, 4, 5} and Y is chosen at random from the subset {1, …, X}, then the joint mass function of X and Y is
b)The conditional mass function of X given that is just
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c)X and Y are not independent. One way of explaining why is to note that for all j, but that for any values of i and j; that is, the conditional probability is not equal to the unconditional probability.
5. Ross 6.42 p. 318
The joint density of X and Y is , where X and Y are positive-valued random variables.
a)To obtain the conditional density of X given , and that of Y, given , we will first need to find the marginal densities of X and Y.
The marginal density of X:
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The marginal density of Y (using integration by parts and then noting that the first term evaluates to 0):
Hence, the conditional density of X given is just
,
and the conditional density of Y given is just
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b)The density function of can be obtained by first determining the cdf and then differentiating with respect to a:
6. Ross 6.45 p. 318
, and we want the probability that the largest is greater than the sum of the others:
(Since the 3 variables are iid, the probability that is larger than the sum of the other two is the same for all i, so we can just examine the case where is largest and then multiply the result by 3).
7. Ross 6.46 p. 318
Since the machine needs at least 3 motors to operate, the length of time that the machine functions is just the length of time the 3rd-largest (or 3rd-smallest in this case) motor functions. I.e., if we look at the order statistics , , , ,and , then the density function of the length of time that the machine functions is just the density function of . Then, using Equation (6.2), the density function of is just
8. Ross 6.57 p. 319
We have that and . Solving these for and gives us and . Also,
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With and being independent random variables, their joint density is just . Hence, from Equation (7.1), we have
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