Question 1

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.

Accepted Answer

The answer of An official from the Australian Securities and...

Question 2

An analysis of the stock market produces the following information about the returns of two stocks: Assume that the returns are positively correlated, with 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.

Accepted Answer

The answer of An analysis of the stock market produces...

Question 3

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?

Accepted Answer

The answer of Let X be a binomial random variable...

Question 4

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

Accepted Answer

The answer of Let X be a Poisson random variable...

Question 5

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?

Accepted Answer

The answer of A market researcher selects 20 students at...

Question 6

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?

Accepted Answer

The answer of The proprietor of a small hardware store...

Question 7

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

Accepted Answer

The answer of Let X be a Poisson random variable...

Question 8

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?

Accepted Answer

The answer of Historical data collected at the Commonwealth Bank...

Question 9

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).

Accepted Answer

The answer of Given a binomial random variable with n...

Question 10

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.

Accepted Answer

The answer of The number of arrivals at a local...

Question 11

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?

Accepted Answer

The answer of Phone calls arrive at the rate of...

Question 12

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.

Accepted Answer

The answer of Consider a binomial random variable X with...

Question 13

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.

Accepted Answer

The answer of An advertising executive receives an average of...

Question 14

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 $$\geq$$ 9.
c. X $$\leq$$ 10.

Accepted Answer

The answer of Given a binomial random variable with n...

Question 15

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. P(4 < X < 9).
c. P(X = 2).
d. P(3 $\le$ X $\le$ 6).

Accepted Answer

The answer of A recent survey in Victoria revealed that...

Quiz 7: Random Variables and Discrete Probability Distributions

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

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

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).

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.

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 $\geq$ 9. c. X $\leq$ 10.

Quiz 7: Random Variables and Discrete Probability Distributions

Let X be a Poisson random variable with μ\muμ = 6. Use the table of Poisson probabilities to find: a. P(X ≤\le≤ 8) b. P(X = 8) c. P(X ≤\le≤ 5) d. P(6 ≤\le≤ X ≤\le≤ 10)

Let X be a Poisson random variable with μ\muμ = 8. Use the table of Poisson probabilities to find: a. P(X ≤\le≤ 6). b. P(X = 4). c. P(X ≥\ge≥ 3). d. P(9 ≤\le≤ X ≤\le≤ 14).

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).

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.

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 ≥\geq≥ 9. c. X ≤\leq≤ 10.

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

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

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).

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 $\geq$ 9. c. X $\leq$ 10.