Suppose that X~ beta (10, 0.8). Calculate P(X = 2) P(X notequalto 2) P(X 5) EX V

Question

Suppose that X~ beta (10, 0.8). Calculate P(X = 2) P(X notequalto 2) P(X 5) EX Var X An auction house offers items of jewelry for sale at its public auctions. Each item has a reserve price which is less than the lower price estimate which, in turn, is less than the upper price estimate. The outcome for any item is independent of the outcomes for all other items. The auction house has found, from past records, the following probabilities for the outcomes of items of jewelry offered for sale. For example, the probability that an item achieves at least its lower price estimate but not its upper price estimate is 0.325. A particular auction includes exactly 40 items of jewelry that may be assumed to be a random sample of such items. Use binomial distributions to find the probability that: at most 10 items, do not achieve their reserve prices; 25 or more items achieve at least their lower price estimates; exactly 2 items achieve at least their upper price estimates; more than 10 items but fewer than 15 items achieve at least their reserve prices but not their lower price estimates. How many of the 40 items of jewelry would you expect to achieve at least their reserve prices but not their upper price estimates?

Explanation / Answer

9.

Given params of binomial distribution:

n=10
p = .8

a. P(X=2) = 10C2(.8^2)(1-.8)^8 = 7.37*E-05
b. P(X!=2) = 1-7.37*E-05 = .99926
c. P(X<3) = P(X=0)+..P(X=2) = 7.79264E-05
d. P(X<=6) = 0.1209
e. P(X>=7) = 0.8791
f. P(X>5) = 0.9672
g. E(X) = np = 10*.8 = 8
h. Var(X) = np(1-p) = 1.6

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