# Mathematics

Statistical Inference I: J. Lee Assignment 3

Problem 1. Approximately 80,000 marriages took place in the state of Pennsylvania last year. Estimate the probability that, for at least one of the couples married in PA last year,

(a) both partners were born on April 30;

(b) both partners celebrated their birthday on the same day of the year.

Make sure to state what assumptions you are making in your computation.

Problem 2. A certain typing agency employs Al, Bob, and Cathy as typists. The average number of errors per page is 3 when typed by Al, 4.2 when typed by Bob, and 2.1 when typed by Cathy. If your 7-page article is equally likely to be typed by any of the three typists, estimate the probability that it will have no errors. Also estimate the probability it will have at most 3 errors.

Problem 3. Let X denote the lifetime (in hours) of a light bulb, and assume that the density function of X is given by

f(x) =

 2x if 0 ≤ x < 1/23/4 if 2 < x < 3 0 otherwise.

(a) On average, what fraction of light bulbs last more than 15 minutes?

(b) Compute E(X).

(c) Compute P (0.25 < X ≤ 2.2 | X > 1).

(d) Compute P (X = 2), P (X = 0), P (X = E(X)).

Problem 4. The density function of X is given by

f(x) =

{ a + bx2 if 0 ≤ x ≤ 1 0 otherwise.

Suppose also that you are told that E(X) = 3/5.

(a) Find a and b.

(b) Determine the cdf, F (x), explicilty.

Problem 5. Define the function F : R → R by

F (x) =

 0 if x < 0 x/2 if 0 ≤ x < 1 (x + 2)/6 if 1 ≤ x < 4 1 if x ≥ 4.

(a) Is F the cdf of a continuous random variable? Explain your answer.

(b) If your answer to part (a) is “yes”, determine the corresponding pdf; if your answer was “no”, then make a modification to F so that it is a cdf, and then compute the corresponding pdf.

(c) Compute the expectation of X, if X has density given by the pdf from part (b).

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