In a quarter-pound bag of red pistachio nuts, some shells are too difficult to pry open by

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Question:

In a quarter-pound bag of red pistachio nuts, some shells are too difficult to pry open by hand. Suppose the random variable X, the number of pistachios in a randomly selected bag that cannot be opened by hand, has the probability distribution given in the table below. x 0 1 2 3 4 5 p(x) 0.600 0.150 0.100 0.075 0.050 0.025

a) Is this probability distribution legitimate? Suppose a cabinet is selected at random.

b) What is the probability that the none of the shells are too difficult to open? No work is required.

c) What is the probability that at most two of the shells are too difficult to open by hand?

d) Find the mean of X.

e) Find the probability X takes on a value less than the mean.

f) Find the variance and standard deviation of X.

g) Suppose the random variable Y is defined by Y = 3X - 2. Find the mean of Y.

h) Find the variance and standard deviation of Y.

i) Suppose the random variable W is defined by W = X 2 - 2. Find the mean of W.

j) Suppose two bags of pistachios are selected at random. What is the probability that both bags have four or more pistachios that are too difficult to open by hand?