stats510 - Homework 3 - Solved

#1
Compute the skewness for the random variable with pdf fX(x) = e−x,x ≥ 0, a pdf that is skewed to the right.

#2
Construct a random right triangle as follows. Let X be a random variable whose distribution is uniform on (0, ). For each X, select a point (d,y) in Quadrant I of the plane with terminal angle X (that is, the angle between the horizontal axis and the line joining the origin with (d,y)) and vertical line segment at y. Here, Y is the height of the random triangle. For a fixed constant d > 0, write the pdf of Y and E[Y ].

#3
This exercise concerns the median of a distribution. Assume that X is continuous.

(a)   Show that E|X − a| is minimized at a = m where m is the median of X. Recall that the median of a random variable X is the value such that P(X ≤ m) = 1/2.

(b)   Let m(X) denote the median of X. Explain why

E(X − m(X))2 ≥ Var[X].

#4
This exercise concerns the Type I Pareto distribution. For any positive real numbers x,α > 0, we say that Y is Pareto(x,α) if it has pdf

 

For each integer n ≥ 1, determine when E[Y n] exists and compute where applicable. Also, determine when the skewness of Y exists and compute it. Explain whether the mgf exists.

#5
Consider the random variable  . Compute E[Y ] when (i) X is a binomial distribution with parameters n ≥ 1 and p ∈ (0,1) and (ii) X is a Poisson distribution with λ > 0. Explain the relationship between your answers.

#6
The log-normal distribution from the example on nonunique moments has an interesting property. If we have the pdf

 ,

then we showed that all the moments exist and are finite. Prove that this distribution does not have an mgf, i.e.

 

does not even exist for any t > 0.

#7
This exercise concerns maximizing expected utility. A common assumption in gambling is that there is a utility function U and when the person must choose between any two gambles X and Y , they prefer X to Y if E[U(X)] > E[U(Y )] and will be indifferent between X and Y if E[U(X)] = E[U(Y )].

(a)   A decision maker has a utility function of the form

U(x) = xc, x ≥ 0.

This person has two choices. They may either earn 1 dollar guaranteed or use that dollar to enter a lottery that pays 500 with probability 0.001 and pays 0 with probability 0.999. For what values of c does the person prefer to play the lottery?

(b)  A decision maker has the same utility function as before, but the gameis as follows. They may either earn Z dollars guaranteed or use that money to take a gamble whose payout is as follows. Flip a fair coin repeatedly until the time T when the first tail appears. The payout is then P(T) = 2T−1 for all T ≥ 1 (note T − 1 is the number of heads, not the number of flips). For what values of c and Z does this person prefer to take the gamble?

#8
Suppose X is a continuous random variable with cdf FX and suppose M(c) = E[ecX] exists for some distinguished c ∈ R. Now define a random variable Y with the cdf

 

(a)   Write the distribution of Y when X has the exponential distribution with parameter λ > 0.

(b)   Under what conditions is the mgf of Y defined in a neighborhood of 0?

Under these conditions, write the mgf of Y .