Question 1

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\end{array} \right.$$ The bonuses are withdrawn from and the fines are charged to the team's account.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?

Accepted Answer

$-1386

Question 2

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.

Accepted Answer

.0025

Question 3

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.

Accepted Answer

P(X=0)=1/40, P(X=1)=15/40, P(X=2)=9/40, P(X=3)=0, P(X=4)=6/40, P(X=5)=9/40.

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.

Accepted Answer

The answer of There are 10 different accounts under study...

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.

Accepted Answer

The answer of Consider the following function: \[f ( x...

Question 6

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 $E \left( 2 ^ { X } - X \right)$ .

Accepted Answer

The answer of Suppose you possess a loaded 6-sided die...

Question 7

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?

Accepted Answer

The answer of Of US citizens, approximately 12% have traveled...

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 .

Accepted Answer

The answer of Busses arrive at a certain stop such...

Question 9

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.

Accepted Answer

The answer of You pick a card at random and...

Question 10

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 $E \left( X ^ { 3 } \right)$

Accepted Answer

The answer of Let be a discrete random variable with...

Question 11

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

Accepted Answer

The answer of Suppose that X is a random variable...

Question 12

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.

Accepted Answer

The answer of Let each letter of the alphabet have...

Question 13

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?

Accepted Answer

The answer of A drawer contains 12 pairs of long...

Question 14

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.

Accepted Answer

The answer of Giant squids have one offspring at a...

Exam 4: Distribution Functions and Discrete Random Variables

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.

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.

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 $E \left( 2 ^ { X } - X \right)$ .

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?

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.

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 $E \left( X ^ { 3 } \right)$

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

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?

Exam 4: Distribution Functions and Discrete Random Variables

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.

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.

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(2X−X)E \left( 2 ^ { X } - X \right)E(2X−X) .

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?

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.

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

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?

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 $E \left( 2 ^ { X } - X \right)$ .