# hw5 - 3 A random sample of 2400 people are asked if they...

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Math 310-1 (Fall 2011) Homework #5 Due date: Wednesday 11/2 in class 1. Exactly one of six similar keys opens a certain door. If you try the keys, one after another, what is the expected number of keys that you will have to try until success? 2. You are oﬀered the following game to play: a fair coin is tossed until heads turns up for the ﬁrst time. If this occurs on the ﬁrst toss you receive 2 dollars, if it occurs on the second toss you receive 2 2 = 4 dollars, and, in general, if heads turns up for the ﬁrst time on the n th toss you receive 2 n dollars. (a) Show that the expected value of your winnings does not exist for this game. (b) Assume that you only receive 2 10 dollars if any number greater than or equal to ten tosses are required to obtain the ﬁrst head. Show that your expected value for this modiﬁed game is ﬁnite and ﬁnd its value.
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Unformatted text preview: 3. A random sample of 2400 people are asked if they favor a government proposal to develop new nuclear power plants. If 40% of the people in the country are in favor of this proposal, ﬁnd the expected value and the standard deviation for the number S 2400 of people in the sample who favor the proposal. 4. Let X be a random variable with E ( X ) = μ and V ar ( X ) = σ 2 . Deﬁne X * = X-μ σ . The random variable X * is called the standardized random variable associated with X . Show that this standardized random variable has expectation 0 and variance 1. From the textbook: pp 104 # 1, 2, 4, 6, 7, 8, 10, 11, 12, 14, 16. (Note that the textbook deﬁnes the geometric distribution as the distribution of “the number of failures before the ﬁrst success in a sequence of Bernoulli trials”.)...
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