Question 1

A person is trying to decide which of two possible mutual funds to invest his money in. Let the random variable X represent the annual return for mutual fund A and let the random variable Y represent the annual return for fund B. It is known that the mean, µ, of X is 10.3% and the standard deviation, Ϭ, of X is 4.2%. It is also known that the mean, µ, of Y is 11.3% and the standard deviation, Ϭ, of Y is 7.2%. Which fund do you think the person would prefer if he is a short-term investor? Which fund do you think he would prefer if he is a long-term investor? Explain your thinking.

Accepted Answer

A short-term investor would probably prefer fund A as it is less volatile (the standard deviation of the returns is smaller). A long-term investor would probably prefer fund B as it has a higher mean return. For a long-term investor the volatility of fund B poses less of a risk.

Question 2

Identify each of the variables in the Binomial Probability Formula. 
$P ( x ) = \frac { n ! } { ( n - x ) ! x ! } \cdot p ^ { x } \cdot ( 1 - p ) ^ { n - x }$ Also, explain what the fraction $\frac { n ! } { ( n - x ) ! x ! }$ computes.

Accepted Answer

n is the fixed number of trials, x is the number of successes, p is the probability of success in one of the n trials, and (1 - p)is the probability of failure in one of the n trials. The fraction determines the number of different orders of x successes out of n trials.

Question 3

The number of lightning strikes in a year at the top of a particular mountain has a Poisson distribution with parameter $\lambda = 3.5$ . Construct a histogram of the probabilities when the number of strikes is from 1-5.

Accepted Answer

The answer of The number of lightning strikes in a...

Question 4

A coin is biased so that the probability it will come up tails is 0.43. The coin is tossed three times. Considering a success to be tails, formulate the process of observing the outcome of the three tosses as a sequence of three Bernoulli trials. Complete the table below by showing each possible outcome together with its probability. Display the probabilities to three decimal places. List the outcomes in which exactly two of the three tosses are tails. Without using the binomial probability formula, find the probability that exactly two of the three tosses are tails. 
$\begin{array} { r | r }  \text { Outcome } & \text { Probability } \ \hline \operatorname { hhh } & ( 0.57 ) ( 0.57 ) ( 0.57 ) = 0.185 \end{array}$

Accepted Answer

The answer of A coin is biased so that the...

Question 5

Five cards are drawn at random, with replacement, from an ordinary deck of 52 cards. Considering success to be drawing a heart, formulate the process of observing the suits of the five cards as a sequence of five Bernoulli trials.

Accepted Answer

The answer of Five cards are drawn at random, with...

Question 6

A game is said to be "fair" if the expected value for winnings is 0, that is, in the long run, the player can expect to win 0. Consider the following game. The game costs $1 to play and the payoffs are $5 for red, $3 for blue, $2 for yellow, and nothing for white. The following probabilities apply. What are your expected winnings? Does the game favor the player or the owner? $$\begin{array} { r | r }  \text { Outcome } & \text { Probability } \ \hline \text { Red } & .02 \ \text { Blue } & .04 \ \text { Yellow } & .16 \ \text { White } & .78 \end{array}$$

Accepted Answer

The answer of A game is said to be "fair"...

Question 7

Explain how you would construct a probability histogram of a discrete random variable given its probability distribution.

Accepted Answer

The answer of Explain how you would construct a probability...

Question 8

Suppose that the random variable X has a binomial distribution and that the success probability, p, is greater than 0.5. Is the probability distribution of X right skewed, left skewed, or symmetric? Explain your thinking.

Accepted Answer

The answer of Suppose that the random variable X has...

Question 9

Suppose a mathematician computed the expected value of winnings for a person playing each of seven different games in a casino. What would you expect to be true for all expected values for these seven games?

Accepted Answer

The answer of Suppose a mathematician computed the expected value...

Question 10

A group of potential jurors consists of 15 women and 18 men. Suppose that 12 people are picked at random from this group, without replacement. Let X represent the number of women among those selected. Since the sample size exceeds 5% of the population size, X does not have an approximate binomial distribution. Explain in your own words why X does not have a binomial distribution. Which of the requirements for a binomial distribution does it not satisfy?

Accepted Answer

The answer of A group of potential jurors consists of...

Question 11

40% of the adult residents of a certain city own their own home. Four residents are selected at random from the city and asked whether or not they own their own home. Considering a success to be "owns their own home", formulate the process of observing whether each of the four residents owns their own home as a sequence of four Bernoulli trials. Complete the table below by showing each possible outcome together with its probability. Display the probabilities to three decimal places. List the outcomes in which exactly two of the four residents own their own home. Without using the binomial probability formula, find the probability that exactly two of the four residents own their own home. $$\begin{array} { r | r }  \text { Outcome } & \text { Probability } \ \hline \operatorname { ssss } ( 0.4 ) ( 0.4 ) ( 0.4 ) ( 0.4 ) = 0.026 \end{array}$$

Accepted Answer

The answer of 40% of the adult residents of a...

Question 12

Give an example of a discrete random variable whose possible values form a countable infinite set of numbers.

Accepted Answer

The answer of Give an example of a discrete random...

Question 13

6.2% of VCRs of a certain type are defective. Let the random variable X represent the number of defective VCRs among 200 randomly selected VCRs of this type. Suppose you wish to find the probability that X is equal to 8. Does the random variable X have a binomial or a Poisson distribution? How can you tell? If X has a binomial distribution, would it be reasonable to use the Poisson approximation? If not, why not?

Accepted Answer

The answer of 6.2% of VCRs of a certain type...

Question 14

The random variable X represents the number of thunderstorms occurring in a month in one city. Suppose that X has a Poisson distribution with parameter $\lambda$ = 3.2. Determine and interpret the mean of the random variable X.

Accepted Answer

The answer of The random variable X represents the number...

Question 15

Let the random variable $X$ represent the winnings at one play of game A. The mean, $\mu$, of $X$ is known to be $- \$ 0.42$ and its standard deviation, $\sigma$, is $\$ 0.27$. Let the random variable $Y$ represent the winnings at one play of game $B$. The mean, $\mu$, of $Y$ is known to be $- \$ 0.42$ and its standard deviation, $\sigma$, is $\$ 0.20$. You have decided to play one of these two games just once. At which game are you more likely to make a profit (i.e., to not lose money)? Explain your thinking.

Accepted Answer

The answer of Let the random variable $X$ represent the...

Question 16

Explain in your own words the meaning of the term "probability distribution".

Accepted Answer

The answer of Explain in your own words the meaning...

Question 17

Let the random variable X represent the winnings at one play of a particular game. The expected value of X is known to be -$0.32. Suppose a player plays the game five times and calculates his average winnings. Will the average definitely be equal to -$0.32? Now suppose the player plays the game 100 times and calculates his average winnings. Will the average definitely be equal to -$0.32? Which average is likely to be closer to -$0.32? Explain your answer with reference to the law of large numbers.

Accepted Answer

The answer of Let the random variable X represent the...

Question 18

For a particular game at a casino, let the random variable X represent the winnings (payoff minus bet)for one play of the game. The expected value of the random variable X is -$0.87. How would you interpret this statement?

Accepted Answer

The answer of For a particular game at a casino,...

Question 19

16% of the employees of a certain company cycle to work. Three employees are selected at random from the company and asked whether or not they cycle to work. Considering a success to be "cycles to work", formulate the process of observing whether each of the three employees cycles to work as a sequence of three Bernoulli trials. Complete the table below by showing each possible outcome together with its probability. Display the probabilities to three decimal places. List the outcomes in which exactly one of the three employees cycles to work. Without using the binomial probability formula, find the probability that exactly one of the three employees cycles to work. $$\begin{array} { r | r }  \text { Outcome } & \text { Probability } \ \hline \text { sss } & ( 0.16 ) ( 0.16 ) ( 0.16 ) = 0.004 \end{array}$$

Accepted Answer

The answer of 16% of the employees of a certain...

Question 20

Which of the following describes the possible values of a Poisson random variable, X?

Accepted Answer

A Poisson random variable can take on any nonnegative integer value (0, 1, 2, 3, ...), representing the number of events occurring in a fixed interval of time or space.

Quiz 5: Discrete Random Variables

The number of lightning strikes in a year at the top of a particular mountain has a Poisson distribution with parameter λ=3.5\lambda = 3.5λ=3.5 . Construct a histogram of the probabilities when the number of strikes is from 1-5.