statistics

Statistical Inference I: J. Lee Assignment 2

Problem 1. In your pocket, you have 1 dime, 2 nickels, and 2 pennies. You select 2 coins at random (without replacement). Let X represent the amount (in cents) that you select from your pocket.

(a) Give (explicitly) the probability mass function for X. Also show a plot of it.

(b) Give (explicitly) the cdf, F(x), for X. Also show a plot of it.

(c) How much money do you expect to draw from your pocket?

Problem 2. Consider a random variable X whose distribution function (cdf) is given by

FX(x) =

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0 if x < �2 0.1 if � 2  x < 1.1 0.3 if 1.1  x < 2 0.6 if 2  x < 3 1 if x � 3.

(a) Give the probability mass function, p(x), of X, explicitly.

(b) Compute P(2 < X < 3).

(c) Compute P(X � 3).

(d) Compute P(X � 3 | X � 0).

(e) What is the cdf (distribution function) of Y = X2? (be explicit!)

(f) Compute E(X). Also compute E(X3 � cos ⇡X).

Problem 3. Six men and five women apply for a job at Alpha, Inc. Three of the applicants are selected (at random) for interviews. Let X denote the number of women in the interview pool.

(a) Give the probability mass function, p(x), of X, explicitly.

(b) Find the probability that either one or two women are in the interview pool.

(c) How many women do you expect to be in the interview pool?

Problem 4. Consider a random variable X whose probability mass function (pmf) is given by

p(x) =

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p if x = �1.9 0.1 if x = �0.1 0.3 if x = 20p p if x = 3 4p if x = 4 0 otherwise.

(a) What is p?

(b) Compute P(1.9  |X|  3).

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(c) What is F(0)? What is F(2)? What is F(F(3.1))? (Here, F(·) denotes the distribution function (cdf) for X)

(d) Sketch a plot of the function F(x). (Make sure to label the coordinates on the axes!)

(e) What is P(2X � 3  4 | X � 2.0)?

(f) Compute E(X).

(g) Compute E(F(X)).

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