Deck 4: Distribution Functions and Discrete Random Variables

Question 1

You pick a card at random and with replacement twice from 3 cards numbered 1, 2, 3. Let S denote their sum and D the result of the first roll minus the second roll. Give the probability mass function of the product of S and D.

Answer 1

$\left( \begin{array} { c c c c c c c } 0 & - 3 & - 5 & - 8 & 3 & 5 & 8 \\3 / 9 & 1 / 9 & 1 / 9 & 1 / 9 & 1 / 9 & 1 / 9 & 1 / 9\end{array} \right)$

Question 2

Let each letter of the alphabet have numerical value equal to its position; i.e., A=1, B=2, , Z=26. A 3-letter word is constructed at random from the letters A, B, C, D, E (any order of letters counts, nonsense words are acceptable, letters cannot be repeated). You then add the numerical values of the letters. Find the expected value of the word.

Answer 2

9

Question 3

Of US citizens, approximately 12% have traveled internationally. A current company wants to hire a new employee who has traveled internationally. How many applicants do they need to interview to have a 60% chance that at least one of the applicants has traveled internationally?

Answer 3

8

Question 4

There are 10 different accounts under study at a local credit union. 3 have $10,000 in the account, 2 have $12,500, 4 have $15,000, and 1 has $20,000. An account is selected at random. Find the expected value and the variance of the money in the account.

Question 5

Consider the following function: $$f ( x ) = \left\{ \begin{array} { l l } 0 & x \leq 0 \\\frac { x ^ { 3 } } { 4 } + c & 0 \leq x \leq 2 \\0 & x > 2\end{array} \right.$$
(a) Find the value of so that is a probability density function.
(b) Let be a random variable with this probability density function. Find the probability that is between 1 and 1.5.
(c) Find the probability that the function $g ( X ) = X ^ { 2 } - 2 X$ is increasing.

Question 6

Giant squids have one offspring at a time until they have a male offspring, at which point they quit reproducing. If they never have a male, they stop after 4 female offspring. They have male offspring with probability .25. Let X be the number of female offspring for a giant squid. Find the probability mass function and the distribution function of X.

Question 7

Let be a discrete random variable with probability mass function: $$\begin{array} { | c | c | c | c | c | } \hline \mathrm { x } & - 1 & 1 & 2 & 3 \\\hline \mathrm { p } ( \mathrm { x } ) & .2 & .3 & .15 & .35 \\\hline\end{array}$$ Find $E \left( X ^ { 3 } \right)$

Question 8

Busses arrive at a certain stop such that in a time interval of length the number of arrivals is a random variable with probability mass function $$f ( i ) = \left\{ \begin{array} { l l } e ^ { - t } \frac { \left( t ^ { i } \right) } { i ! } & i = 0,1,2 , \ldots \\0 & \text { elsewhere }\end{array} \right.$$
(a) Prove that is indeed a probability mass function.
(b) Find the probability of at least 3 arrivals in one time period of length .

Question 9

A drawer contains 12 pairs of long socks and 10 pairs of short socks. If a traveler selects 4 pairs of socks at random to pack, what is the probability mass function of the number of pairs of long socks?

Question 10

Suppose that X is a random variable such that E((X-2)(X+2))=12, Var(2X+3)=28. Find E(X).

Question 11

The number of finishers at a particularly difficult race is a random variable with probability mass function: $$f ( x ) = \left\{ \begin{array} { l l } \frac { 36 } { 35 ( x ) ( x + 1 ) } & x = 1 , \ldots , 35 \\0 & \text { else. }\end{array} \right.$$ Find the probability that at least 3 people do not finish.

Question 12

Suppose you possess a loaded 6-sided die in which 1 is twice as likely as each of the other 5 outcomes. Let X be the outcome of rolling this die once. Find $E \left( 2 ^ { X } - X \right)$ .

Question 13

On a certain athletics team, athletes get a $250 bonus if they run their one-mile race in under 4 minutes. If they do not, they pay the club $10. There are 7 athletes on the team. The number of athletes who run their mile in under 4 minutes has the probability mass function. $$f ( x ) = \left\{ \begin{array} { l l } \frac { x ^ { 2 } } { 140 } & x = 0,1 , \ldots , 22 \\0 & \text { else. }\end{array} \right.$$ The bonuses are withdrawn from and the fines are charged to the team's account. After all the athletes have run their mile, what is the expected value of the change in the balance of the team's account?

Question 14

Let X be the number of cars belonging to a random family in Beverly Hills. Suppose that X has distribution function F(x), where F(0)=1/40, F(1)=16/40, F(2)=25/40), F(3)=25/40, F(4)=31/40, also suppose P(X=5)=P(X>5). Find P(X=i) for i=0,…,5.