Deck 7: Random Variables and Discrete Probability Distributions

Question 1

A discrete random variable can take either finite or infinite values as long as the values are countable.

Answer 1

True

Question 2

A Poisson distribution with $\mu$ = .60 is a:

A)symmetrical distribution.
B)negatively skewed distribution (skewed to the left).
C)positively skewed distribution (skewed to the right).
D)binomial distribution.

Answer 2

positively skewed distribution (skewed to the right).

Question 3

Gender is an example of a continuous random variable.

Answer 3

False

Question 4

If X and Y are any random variables with E(X)= 3, E(Y) = 2, E(XY) = 12, V(X) = 16 and V(Y) = 25, then the relationship between X and Y is a: $$\text { Hint: } \operatorname { corr } ( X , Y ) = \operatorname { Cov } ( X , Y ) / ( ( \sqrt { V ( X ) } \sqrt { V ( Y ) } ) = ( E ( X Y ) - E ( X ) E ( Y ) ) / ( \sqrt { V ( X ) } \sqrt { V ( Y ) } )$$

A)weak negative relationship.
B)strong positive relationship.
C)strong negative relationship.
D)weak positive relationship.

Question 5

The lottery commission has designed a new instant lottery game. Players pay $1.00 to scratch a ticket, where the prize won, X, (measured in $) has the following discrete probability distribution : $$\begin{array} { | c | c | } \hline X & P [ X ] \\\hline 0 & 0.95 \\\hline 10 & 0.049 \\\hline 100 & 0.001 \\\hline\end{array}$$ Which of the following best describes the expected value of X ?

A)In the long run, the average prize won per $1 played is $0.
B)In the long run, the average prize won per $1 played is $0.41.
C)In the long run, the average prize won per $1 played is $0.59.
D)None of these choices are correct

Question 6

Which of the following cannot generate a Poisson distribution?

A)The number of children watching a movie.
B)The number of telephone calls received by a switchboard in a specified time period.
C)The number of customers arriving at a petrol station on Christmas day.
D)The number of bacteria found in a cubic yard of soil.

Question 7

Which probability distribution is appropriate when the events of interest occur randomly, independently of one another, and rarely?

A)Binomial distribution.
B)Poisson distribution.
C)Any discrete distribution.
D)Any continuous distribution.

Question 8

For a discrete probability distribution to be valid, the probabilities must lie between 0 and 1, where the sum of all probabilities must be 1.

Question 9

Given that X is a binomial random variable, the binomial probability P(X $\geq$ x) is approximated by the area under a normal curve to the right of:

A)x - 0.5.
B)x + 0.5.
C)x - 1.
D)x + 1.

Question 10

A binomial distribution for which the number of trials n is large can well be approximated by a Poisson distribution when the probability of success, p, is:

A)larger than 0.95.
B)larger than 0.50.
C)between 0.25 and 0.50.
D)smaller than 0.05.

Question 11

Which of the following is not a characteristic of a binomial experiment?

A)There is a sequence of identical trials.
B)Each trial results in two or more outcomes.
C)The trials are independent of each other.
D)Probability of success p is the same from one trial to another.

Question 12

A table, formula, or graph that shows all possible countable values a random variable can assume, together with their associated probabilities, is called a:

A)discrete probability distribution.
B)continuous probability distribution.
C)bivariate probability distribution.
D)probability tree.

Question 13

The Poisson random variable is a:

A)discrete random variable with infinitely many possible values.
B)discrete random variable with a finite number of possible values.
C)continuous random variable with infinitely many possible values.
D)continuous random variable with a finite number of possible values.

Question 14

Which of the following best describes a function that assigns a numerical value to each simple event in a sample space?

A)An expected value.
B)The mean.
C)A random variable.
D)The standard deviation.

Question 15

The standard deviation of a binomial distribution for which n = 100 and p = .35 is:

A)4.77.
B)2.275.
C)47.7.
D)22.75.

Question 16

The lottery commission has designed a new instant lottery game. Players pay $1.00 to scratch a ticket, where the prize won, X, (measured in $) has the following discrete probability distribution : $$\begin{array} { | c | c | } \hline X & P [ X ] \\\hline 0 & 0.95 \\\hline 10 & 0.049 \\\hline 100 & 0.001 \\\hline\end{array}$$ Which of the following best describes the standard deviation of X ?

A)14.552 in $2
B)3.815 in $
C)0.348 in $
D)None of these choices are correct

Question 17

The number of accidents that occur annually on a busy stretch of highway is an example of:

A)a discrete random variable.
B)a continuous random variable.
C)a discrete probability distribution.
D)a continuous probability distribution.

Question 18

The expected value, E(X), of a binomial probability distribution is:

A)n + p.
B)np(1 - p).
C)np.
D)n + p - 1.

Question 19

The weighted average of the possible values that a random variable X can assume, where the weights are the probabilities of occurrence of those values, is referred to as the:

A)variance.
B)standard deviation.
C)expected value.
D)covariance.

Question 20

Which of the following best describes a discrete random variable?

A)A random variable that can only assume a countable number of values.
B)A random variable that can only assume an uncountable number of values.
C)All random variable are discrete random variables
D)None of these choices are correct

Question 21

Let X represent the number of computers in an Australian household, for those that own a computer. a. Find and interpret the expected number of computers in a randomly selected Australian household.
b. Find the variance of the number of computers in a randomly selected Australian household.

Question 22

A bivariate distribution is a distribution is a joint probability distribution of two variables.

Question 23

The number of people winning a lottery ticket each week is an example of a Poisson variable.

Question 24

Given a binomial random variable with n = 20 and p = 0.6, find the following probabilities using the binomial table.
a. P(X $\le$ 13).
b. P(X $\ge$ 15).
c. P(X = 17).
d. $P ( 11 < X < 14 )$

Question 25

An analysis of the stock market produces the following information about the returns of two stocks: $\begin{array}{lcc} & \text { Stock } 1 & \text { Stock } 2 \\\hline \text { Expected Returns } & 15 \% & 18 \% \\\text { Standard Deviations } & 20 & 32 \\\hline & &\end{array}$ Assume that the returns are positively correlated, with $\rho$ 12 = 0.80.
a. Find the mean and standard deviation of the return on a portfolio consisting of an equal investment in each of the two stocks.
b. Suppose that you wish to invest $1 million. Discuss whether you should invest your money in stock 1, stock 2, or a portfolio composed of an equal amount of investments on both stocks.

Question 26

Let X and Y be two independent random variables with the following probability distributions:

a Find the probability distribution of the random variable X Y.
b. Check whether E(XY)=E(X) ' E(Y) by separately evaluating each side of the equality.

Question 27

A Bernoulli trial is where each trial of an experiment has four possible outcomes, the probability of success is p and the trials are not independent.

Question 28

Consider a binomial random variable X with n = 7 and p = 0.3.
a. Find the probability distribution of X.
b. Find P(X < 3).
c. Find the mean and the variance of X.

Question 29

For each of the following random variables, indicate whether the variable is discrete or continuous, and specify the possible values that it can assume.
a. X = The number of animals visited by a veterinarian in one day.
b. X = Closing share price of a particular stock over one month.
c. X = The weights of new members at a gymnasium.
d. X = The number of students attending a lecture, where the theatre seats 250 people.
e. X = The temperature at a seaside resort in the summer.

Question 30

Let X and Y be two independent random variables with the following probability distributions:

Question 31

An official from the Australian Securities and Investments Commission estimates that 75% of all investment bankers have profited from the use of insider information. If 15 investment bankers are selected at random from the Commission's registry, find the probability that:
a. at most 10 have profited from insider information.
b. at least six have profited from insider information.
c. all 15 have profited from insider information.

Question 32

A recent survey in Victoria revealed that 60% of the vehicles travelling on highways, where speed limits are posted at 100 kilometres per hour, were exceeding the limit. Suppose you randomly record the speeds of 10 vehicles travelling on the Hume Highway, where the speed limit is 100 kilometres per hour. Let X denote the number of vehicles that were exceeding the limit. Find the following probabilities.
a. P(X=10).
b.
C. P(X=2).
d.

Question 33

The probability distribution for X is as follows: Find the expected value of Y = X + 10.

Question 34

The joint probability distribution of X and Y is shown in the following table.
a. Determine the marginal probability distributions of X and Y.
b. Are X and Y independent? Explain.
c. Find
d. Find the probability distribution of the random variable X+Y.
e. Find E(XY).
f. Find COV(X, Y).

Question 35

The Binomial distribution and the Poisson distribution are discrete distributions.

Question 36

Let X represent the number of computers in Australian households who own computers. The probability distribution of X is as follows: What is the probability that a randomly selected Australian household will have:
a. more than 2 computers?
b. between 2 and 5 computers, inclusive?
c. fewer than 3 computers?

Question 37

State whether or not each of the following are valid probability distributions, and if not, explain why not.
a.
b.
c.

Question 38

The probability distribution for X is as follows: a. Find E[5X + 1].
b. Find V[5X + 1].

Question 39

The P(X ≤ x) is an example of a cumulative probability.

Question 40

The probability distribution for X ,daily demand of a particular newspaper at a local newsagency,( in hundreds) is as follows: $$\begin{array} { l | r r r r } x & 1 & 2 & 3 & 4 \\\hline p ( x ) & 0.05 & 0.42 & 0.44 & 0.09\end{array}$$ a. Find and interpret the expected value of X.
b. Find V(X).
c. Find $\sigma$ .

Question 41

Phone calls arrive at the rate of 30 per hour at the reservation desk for a hotel.
a. Find the probability of receiving two calls in a five-minute interval of time.
b. Find the probability of receiving exactly eight calls in 15 minutes.
c. If no calls are currently being processed, what is the probability that the desk employee can take a four-minute break without being interrupted?

Question 42

The number of arrivals at a local petrol station between 3:00 and 5:00pm has a Poisson distribution with a mean of 12.
a. Find the probability that the number of arrivals between 3:00 and 5:00pm is at least 10.
b. Find the probability that the number of arrivals between 3:30 and 4:00pm is at least 10.
c. Find the probability that the number of arrivals between 4:00 and 5:00pm is exactly two.

Question 43

Let X be a binomial random variable with n = 25 and p = 0.01.
a. Use the binomial table to find P(X = 0), P(X = 1), and P(X = 2).
b. Approximate the three probabilities in part (a) using the appropriate Poisson distribution.
c. Compare your approximations in part (b) with the exact probabilities found in part (a). What is your conclusion?

Question 44

Let X be a Poisson random variable with $\mu$ = 6. Use the table of Poisson probabilities to find:
a. $P ( X < 8 )$
b. $P ( X = 8 )$
c. $P ( X \geq 5 )$
d. $P ( 6 \leq X \leq 10 )$

Question 45

Let X be a Poisson random variable with $\mu$ = 8. Use the table of Poisson probabilities to find:
a. $P ( X \leq 6 )$
b. $P ( X = 4 )$
c. $P ( X \geq 3 )$ .
d. $P ( 9 \leq X \leq 14 )$ .

Question 46

Historical data collected at the Commonwealth Bank in Sydney revealed that 80% of all customers applying for a loan are accepted. Suppose that 50 new loan applications are selected at random.
a. Find the expected value and the standard deviation of the number of loans that will be accepted by the bank.
b. What is the probability that at least 42 loans will be accepted?
c. What is the probability that the number of loans rejected is between 10 and 15, inclusive?

Question 47

Given a binomial random variable with n =15 and p = 0.40, find the exact probabilities of the following events and their normal approximations.
a. X = 6.
b. X 9.
c. X 10.

Question 48

An advertising executive receives an average of 10 telephone calls each afternoon between 2 and 4pm. The calls occur randomly and independently of one another.
a. Find the probability that the executive will receive 13 calls between 2 and 4pm on a particular afternoon.
b. Find the probability that the executive will receive seven calls between 2 and 3pm on a particular afternoon.
c. Find the probability that the executive will receive at least five calls between 2 and 4pm on a particular afternoon.

Question 49

A market researcher selects 20 students at random to participate in a wine-tasting test. Each student is blindfolded and asked to take a drink out of each of two glasses, one containing an expensive wine and the other containing a cheap wine. The students are then asked to identify the more expensive wine. If the students have no ability whatsoever to discern the more expensive wine, what is the probability that the more expensive wine will be correctly identified by:
a. more than half of the students?
b. none of the students?
c. all of the students?
d. eight of the students?

Question 50

The proprietor of a small hardware store employs three men and three women. He will select three employees at random to work on Christmas Eve. Let X represent the number of women selected.
a. Express the probability distribution of X in tabular form.
b. What is the probability that at least two women will work on Christmas Eve?