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Chapter 7: Q.7.31 (page 354)
In Problem 7.6, calculate the variance of the sum of the rolls.
The variance of the sum of the rolls are.
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In the text, we noted that
when the are all nonnegative random variables. Since
an integral is a limit of sums, one might expect that
whenever are all nonnegative random
variables; this result is indeed true. Use it to give another proof of the result that for a nonnegative random variable ,
Hint: Define, for each nonnegative , the random variable
by
role="math" localid="1647348183162"
Now relate
The random variables X and Y have a joint density function is given by
Compute
Show that is stochastically larger than if and only if
for all increasing functions .
Hint: Show that , then by showing that and then using Theoretical Exercise 7.7. To show that if for all increasing functions , then , define an appropriate increasing function .
Consider an urn containing a large number of coins, and suppose that each of the coins has some probability p of turning up heads when it is flipped. However, this value of varies from coin to coin. Suppose that the composition of the urn is such that if a coin is selected at random from it, then the value of the coin can be regarded as being the value of a random variable that is uniformly distributed over . If a coin is selected at random from the urn and flipped twice, compute the probability that
a. The first flip results in a head;
b. both flips result in heads.
Let X be the length of the initial run in a random ordering of n ones and m zeros. That is, if the first k values are the same (either all ones or all zeros), then X 脷 k. Find E[X].
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