STATISTICS 244
Problem Set 5
9:0
010:20 TTh
Due Thursday February 16, 2017
1.
Many nonnormal distributions are approximately normal, N(
μ,
s
2
).
One example is
the Beta(
a
,
b
) distribution, as long as
a
and
b
are moderately large and the Beta
expectation and variance are used for
μ
and
s
2
.
Suppose X has a Beta density.
(i) Use
the normal approximation to find all three of P(X–.6 <.001), P(X–.6 <.01),
and P(X–.6 <.1), when X is (a) Beta(3,2), (b) Beta(30,20), (c) Beta(300, 200). (Present
the results in a 3x3 table.)
(ii) For case (a) only, find the same three probabilities exactly
directly from the beta density by integration.
2.
Election ties.
What is the chance that there will be an exact tie in an election?
Suppose there are n voters and that n is even (otherwise a tie is impossible).
Suppose the
voters vote independently of one another, each with probability p of voting for candidate
A and probability 1 – p of voting for candidate B.
Let X = total vote for A; then there
will be a tie only if X =n/2.
(a) What is the probability of a tie if p = .5 and n = 20?
(b) What is the probability of a tie if p = .6 and n = 20?
(c) Suppose p is unknown, and our uncertainty about it is described adequately by a Beta
(4,4).
What is the probability of a tie (i.e. the marginal probability P(X = n/2)), if n = 20?
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 Fall '08
 DRTON
 Statistics, Normal Distribution

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