91Ó°ÊÓ

Let \(X\) denote the number of times a certain numerical control machine will malfunction: \(1,2,\) or 3 times on any given day. Let \(Y\) denote the number of times a technician is called on an emergency call. Their joint probability distribution is given as $$ \begin{array}{cc|ccc} & & & x & \\ {f(x, y)}& & 1 & 2 & 3 \\ \hline & 1 & 0.05 & 0.05 & 0.1 \\ \text { y } & 2 & 0.05 & 0.1 & 0.35 \\ & 3 & 0 & 0.2 & 0.1 \end{array} $$ (a) Evaluate the marginal distribution of \(X\). (b) Evaluate the marginal distribution of \(Y\). (c) Find \(P(Y=3 \mid X=2).\)

Let \(X\) denote the: number of heads and \(Y\) the number of heads minus the number of tails when 3 coins are tossed. Find the joint probability distribution of \(X\) and \(Y\).

Three cards are drawn without replacement from the 12 face cards (jacks, queens, and kings) of an ordinary deck of 52 playing cards. Let \(X\) be the number of kings selected and \(Y\) the number of jacks. Find (a) the joint probability distribution of \(X\) and \(Y\); (b) \(P[(X, Y)\) e \(A\) ]: where \(A\) is the region given by \(\\{(x, y) \quad \mid x+y>2\\}.\)

A coin is tossed twice. Let \(Z\) denote the number of heads on the first toss and \(W\) the total number of heads on the 2 tosses. If the coin is unbalanced and a head has a \(40 \%\) chance of occurring, find (a) the joint probability distribution of \(W\) and \(Z\); (b) the marginal distribution of \(W\); (c) the marginal distribution of \(Z\); (d) the probability that at least 1 head occurs.

The joint density function of the random variables \(X\) and \(Y\) is $$ f(x, y)=\left\\{\begin{array}{ll} 6 x, & 0< x < 1,0 < y < 1-x, \\ 0, & \text { elsewhere. } \end{array}\right. $$ (a) Show that \(X\) and \(Y\) are not independent. (b) Find \(P(X>0.3 \quad Y=0.5)\).

A tobacco company produces blends of tobacco with each blend containing various proportions of Turkish, domestic, and other tobaccos. The proportions of Turkish and domestic in a blend are random variables with joint density function \((X=\) Turkish and \(Y=\) domestic \()\) $$ f(x, y)=\left\\{\begin{array}{ll} 24 x y, & 0 \leq x, y<_{-} 1: x+y \leq 1, \\ 0, & \text { elsewhere; } \end{array}\right. $$ (a) Find the probability that in a given box the Turkish tobacco accounts for over half the blend. (b) Find the marginal density function for the proportion of the domestic tobacco. (c) Find the probability that the proportion of Turkish tobacco is less than \(1 / 8\) if it is known that the blend contains \(3 / 4\) domestic tobacco.

An insurance company offers its policyholders a number of different premium payment options. For a randomly selected policyholder, let \(X\) be the number of months between successive payments. The cumulative distribution function of \(X\) is $$ F(x)=\left\\{\begin{array}{ll} 0, & \text { if } x<1 \\ 0.4, & \text { if } 1 \leq x<3, \\ 0.6, & \text { if } 3 \leq x<5, \\ 0.8, & \text { if } 5 \leq x<7, \\ 1.0, & \text { if } x \geq 7. \end{array}\right. $$ (a) What is the probability mass function of \(X ?\) (b) Compute \(\mathrm{P}(4

An industrial process manufactures items that can be classified as either defective or not defective. The probability that an item is defective is \(0.1 .\) An experiment is conducted in which 5 items are drawn randomly from the process. Let the random variable \(X\) be the number of defectives in this sample of \(5 .\) What is the probability mass function of \(X ?\)

A chemical system that results from a chemical reaction has two important components among others in a blend. The joint distribution describing the proportion \(X_{1}\) and \(X_{2}\) of these two components is given by $$ f\left(x_{1}, x_{2}\right)=\left\\{\begin{array}{ll} 2, & 0 < x_{1}< x_{2} < 1, \\ (0, & \text { elsewhere. } \end{array}\right. $$ (a) Give the marginal distribution of \(X_{1}\). (b) Give the marginal distribution of \(X_{2}\).(c) What is the probability that component proportions produce the results \(X_{1}<0.2\) and \(X_{2}>0.5 ?\) (d) Give the conditional distribution \(f_{X_{1} \mid X_{2}}\left(x_{1} \mid x_{2}\right)\).