91Ó°ÊÓ

If n houses are located at various points along a straight road, at what point along the road should a store be located in order to minimize the sum of the distances from the n houses to the store?

Let X be a random variable having the binomial distribution with parameters \(n = 7\)and\(p = \frac{1}{4}\), and let Y be a random variable having the binomial distribution with parameters \(n = 5\) and \(p = \frac{1}{2}\). Which of these two random variables can be predicted with the smaller M.S.E?

Show that two random variablesXandYcannot possibly have the following properties:\(E\left( X \right) = 3\),\(E\left( Y \right) = 2\),\(E\left( {{X^2}} \right) = 10\),\(E\left( {{Y^2}} \right) = 29\), and\(E\left( {XY} \right) = 0\).

Suppose thatXis a random variable for which the m.g.f. is as follows:\(\psi \left( t \right) = \frac{1}{5}{e^t} + \frac{2}{5}{e^{4t}} + \frac{2}{5}{e^{8t}}\)for−∞< t <∞.Find the probability distribution ofX. Hint:It is a simple discrete distribution.

Consider a coin for which the probability of obtaining a head on each given toss is 0.3. Suppose that the coin is to be tossed 15 times, and let X denote the number of heads that will be obtained.

1. What prediction of X has the smallest M.S.E?
2. What prediction of X has the smallest M.A.E?

Suppose thatXis a random variable for which the m.g.f. is as follows:\(\psi \left( t \right) = \frac{1}{6}\left( {4 + {e^t} + {e^{ - t}}} \right)\)for−∞< t <∞. Find the probability distribution ofX.

Suppose thatXandYhave a continuous joint distribution

for which the joint p.d.f. is as follows:

\(f\left( {x,y} \right) = \left\{ {\begin{align}{}{\frac{1}{3}\left( {x + y} \right)}&{0 \le x \le 1,0 \le y \le 2}\\0&{otherwise}\end{align}} \right.\)

Determine the value of Var(2X−3Y+8).

Suppose that X and Y are random variables such that

\(E\left( {Y|X} \right) = aX + b\)Assuming that\(Cov\left( {X,Y} \right)\)exists and that\(0 < Var\left( X \right) < \infty \), determine expressions for a and b in terms of\(E\left( X \right)\),\(E\left( Y \right)\)and\(Cov\left( {X,Y} \right)\).

Suppose thatXandYare random variables for whichE(X)=3,E(Y)=1, Var(X)=4, and Var(Y )=9. LetZ=5X−Y+15. FindE(Z)and Var(Z)under each of thefollowing conditions:

(a)XandYare independent;

(b)XandYare uncorrelated;

(c) the correlation ofXandYis 0.25.

Let X have pdf:

\(f\left( x \right) = \left\{ {\begin{aligned}{{}{}}{{x^{ - 2}};{\rm{ if }}x > 1}\\{0;{\rm{ otherwise}}}\end{aligned}} \right.\)

Prove that the m.g.f.\(\psi \left( t \right)\) is finite for all \(t \le 0\) but for no \(t > 0\).