Chapter 4: Random Variables

There are n components lined up in a linear arrangement. Suppose that each component independently functions with probability p. What is the probability that no 2 neighboring components are both nonfunctional?

Each of the members of a $7$-judge panel independently makes a correct decision with probability .$7$. If the panel’s decision is made by majority rule, what is the probability that the panel makes the correct decision? Given that $4$ of the judges agreed, what is the probability that the panel made the correct decision?

Let X be a binomial random variable with parameters (n, p). What value of p maximizes P{X = k}, k = 0, 1, ... , n? This is an example of a statistical method used to estimate p when a binomial (n, p) random variable is observed to equal k. If we assume that n is known, then we estimate p by choosing that value of p that maximizes P{X = k}. This is known as the method of maximum likelihood estimation.

A salesman has scheduled two appointments to sell encyclopedias. His first appointment will lead to a sale with probability $.3,$and his second will lead independently to a sale with probability $.6$. Any sale made is equally likely to be either for the deluxe model, which costs $\backslash (1000$, or the standard model, which costs $\backslash )500.$ Determine the probability mass function of $X$, the total dollar value of all sales

A family has n children with probability $\alpha p^n,n\ge 1$where$\alpha \le (1-p)/p$

(a) What proportion of families has no children?

(b) If each child is equally likely to be a boy or a girl (independently of each other), what proportion of families consists of k boys (and any number of girls)?

On average, $5.2$hurricanes hit a certain region in a year. What is the probability that there will be $3$or fewer hurricanes hitting this year?

Five distinct numbers are randomly distributed to players numbered$1$through $5.$ Whenever two players compare their numbers, the one with the higher one is declared the winner. Initially, players$1$and $2$ compare their numbers; the winner then compares her number with that of player $3,$ and so on. Let $X$ denote the number of times player $1$ is a winner. Find$PX=i,i=0,1,2,3,4.$

Suppose that n independent tosses of a coin having probability p of coming up heads are made. Show that the probability that an even number of heads results is

$\frac{1}{2}\left[1+(q-p{)}^n\right]$.where$q=1-p$.Do this by proving and then utilizing the identity

$\sum _{i=0}^{[n/2]}\left(\begin{array}{c}n\\ 2i\end{array}\right)p^{2i}{q}^{n-2i}=\frac{1}{2}\left[(p+q{)}^n+(q-p{)}^n\right]$

The number of eggs laid on a tree leaf by an insect of a certain type is a Poisson random variable with parameter $\lambda$. However, such a random variable can be observed only if it is positive, since if it is $0$, then we cannot know that such an insect was on the leaf. If we let $Y$denote the observed number of eggs, then

$P\{Y=i\}=P\{X=i\mid X>0\}$

where $X$is Poisson with parameter $\lambda$. Find $E\left[Y\right]$.

The National Basketball Association (NBA) draft lottery involves the 11 teams that had the worst won-lost records during the year. A total of 66 balls are placed in an urn. Each of these balls is inscribed with the name of a team: Eleven have the name of the team with the worst record, 10 have the name of the team with the second worst record, 9 have the name of the team with the third worst record, and so on (with 1 ball having the name of the team with the 11 th-worst record). A ball is then chosen at random, and the team whose name is on the ball is given the first pick in the draft of players about to enter the league. Another ball is then chosen, and if it "belongs" to a team different from the one that received the first draft pick, then the team to which it belongs receives the second draft pick. (If the ball belongs to the team receiving the first pick, then it is discarded and another one is chosen; this continues until the ball of another team is chosen.) Finally, another ball is chosen, and the team named on the ball (provided that it is different from the previous two teams) receives the third draft pick. The remaining draft picks 4 through 11 are then awarded to the 8 teams that did not "win the lottery," in inverse order of their won-lost not receive any of the 3 lottery picks, then that team would receive the fourth draft pick. Let X denote the draft pick of the team with the worst record. Find the probability mass function of X.