Question 1

The Securities and Exchange Commission has determined that the number of companies listed on the NYSE declaring bankruptcy is approximately a Poisson distribution with a mean of 2.6 per month.Find the probability that more than 1 bankruptcy occurs next month.
A).1931
B).9257
C).7326
D).4816
E).2674

Accepted Answer

P(1) = e^−2.6(2.6)¹/1! (e = 2.71828)

Question 2

The Securities and Exchange Commission has determined that the number of companies listed on the NYSE declaring bankruptcy is approximately a Poisson distribution with a mean of 2.6 per month.Find the probability that exactly 4 bankruptcies occur next month.
A).8774
B).1414
C).1557
D).2176

Accepted Answer

We can use the Poisson probability formula to calculate the probability of 4 bankruptcies occurring next month:

P(X=4) = (e^-2.6 * 2.6^4) / 4!

P(X=4) = 0.1414

Therefore, the probability that exactly 4 bankruptcies occur next month is 0.1414, which corresponds to option B).

Question 3

If n = 15 and p = .4,then the standard deviation of the binomial distribution is:
A)9
B)6
C)3.6
D)1.897
E).4

Accepted Answer

The answer of If n = 15 and p =...

Question 4

The requirement that the probability of success remains constant from trial to trial is a property of the _______________ distribution.
A)Binomial
B)Poisson
C)Hypergeometric
D)Continuous

Accepted Answer

The requirement that the probability of success remains constant from trial to trial is the characteristic of the binomial distribution. The Poisson distribution deals with the number of events occurring in a fixed interval of time or space, and the probability of occurrence may not be constant. The hypergeometric distribution deals with sampling without replacement and the probability of success changes from trial to trial. The continuous distribution deals with continuous random variables such as time or height, and not discrete events.

Question 5

If x is a binomial random variable,then the standard deviation of x is given by: A)np B)(npq)² C)√npq D)npq

Accepted Answer

The formula for the standard deviation of a binomial random variable is 𝜎 = √(npq), where n is the number of trials, p is the probability of success on a single trial, and q is the probability of failure on a single trial (q=1-p). Therefore, choice (C) of √(npq) is the correct answer. Choices (A), (B), and (D) are incorrect.

Question 6

A fair die is rolled 10 times.What is the average number of even number outcomes?
A)3
B)4
C)5
D)6
E)7

Accepted Answer

Binomial mean = μ = np = 10(.5) = 5

Question 7

A fair die is rolled 36 times.What is the standard deviation of the even number (2,4,or 6)outcomes?
A)18
B)9
C)5
D)3
E)1.732

Accepted Answer

The probability of rolling an even number (2,4, or 6) on a fair die is 1/2. So, the expected number of even number outcomes in 36 rolls is (1/2)*36 = 18. The variance of the number of even number outcomes is given by Var(X) = npq = (36/2)(1/2)(1/2) = 9. Taking the square root gives us the standard deviation, which is 3. Therefore, the correct answer is (D).

Question 8

The binomial distribution is characterized by situations that are analogous to
A)Drawing balls from an urn.
B)Coin tossing.
C)Counting defects on an item.
D)Measuring the length of an item.

Accepted Answer

The binomial distribution is often used to model situations where there are two possible outcomes (success/failure) and a fixed number of trials. Coin tossing is a classic example of a binomial distribution, where success is getting heads and failure is getting tails. Therefore, B) is the best choice.

Question 9

The equation for the variance of the binomial distribution is given by: A)p^x(1 - p)^n-x B)np C)np(1 - p) D)

Accepted Answer

The formula for the variance of the binomial distribution is np(1-p), where n is the number of trials and p is the probability of success on each trial. This can be derived from the formula for the variance of a Bernoulli distribution (p(1-p)) by multiplying by the number of trials (n). Therefore, the correct choice is C.

Question 10

The Securities and Exchange Commission has determined that the number of companies listed on the NYSE declaring bankruptcy is approximately a Poisson distribution with a mean of 2.6 per month.Find the probability that no more than one bankruptcy occurs next month.
A).1931
B).9257
C).7326
D).4816
E).2674

Accepted Answer

The answer of The Securities and Exchange Commission has determined...

Question 11

A fair die is rolled 10 times.What is the probability that an odd number (1,3,or 5)will occur fewer than 3 times?
A).0547
B).1172
C).1550
D).7752
E).8450

Accepted Answer

First, we need to find the probability of getting an odd number on any single roll, which is 3/6 or 1/2. Then, using the binomial probability formula, we can find the probability of getting an odd number fewer than 3 times in 10 rolls:
P(X<3) = P(X=0) + P(X=1) + P(X=2)
= C(10,0)*(1/2)^0*(1/2)^10 + C(10,1)*(1/2)^1*(1/2)^9 + C(10,2)*(1/2)^2*(1/2)^8
= 0.0547 (rounded to four decimal places)
Therefore, the correct answer is choice A.

Question 12

When p = .5,the binomial distribution will _________ be symmetric.
A)Always
B)Sometimes
C)Never

Accepted Answer

When p = .5, the probability of success and failure is equal, resulting in a symmetric distribution. At extreme values of p (close to 0 or 1), the distribution becomes more skewed.

Question 13

Two characteristics,or assumptions,of the Poisson distribution are:
A)The probability of success remains constant from trial to trial,and the random variable of interest is continuous.
B)The event occurring in one interval is independent of the event occurring in any other nonoverlapping interval,and the random variable of interest is continuous.
C)The event occurring in one interval is independent of the event occurring in any other nonoverlapping interval,and the random variable of interest is discrete.
D)The event occurring in one interval is dependent on the event occurring in any other nonoverlapping interval,and the random variable of interest is continuous.

Accepted Answer

The two characteristics of the Poisson distribution are that the event occurring in one interval is independent of the event occurring in any other nonoverlapping interval, and the random variable of interest is discrete (it counts the number of events that occur in a certain time or space). The probability of success does not need to remain constant, as the Poisson distribution is used to model rare events with a low probability of occurrence.

Question 14

Which of the following statements about the binomial distribution is not correct?
A)Each trial results in a success or failure.
B)Trials are independent of each other.
C)The probability of success remains constant from trial to trial.
D)The random variable of interest is continuous.
E)The experiment consists of n identical trials.

Accepted Answer

The random variable of interest in a binomial distribution is discrete, not continuous. It represents the number of successes in a fixed number of trials.

Question 15

A fair die is rolled 10 times.What is the probability that an even number (2,4,or 6)will occur between 2 and 4 times?
A).6123
B).1709
C).1611
D).3662
E).3223

Accepted Answer

The answer of A fair die is rolled 10 times.What...

Question 16

For a random variable X,the mean value of the squared deviations of its values from their expected value is called its ____________.
A)Standard deviation
B)Mean
C)Probability
D)Variance

Accepted Answer

The mean value of the squared deviations is the definition of variance, denoted as Var(X). The standard deviation is the square root of the variance. The mean and probability are not related to the variance.

Question 17

A random variable that is defined to be the total number of successes in n trials is a __________ random variable.
A)Binomial
B)Poisson
C)Hypergeometric
D)Continuous

Accepted Answer

A binomial random variable is defined as the total number of successes in n independent and identical trials, where each trial has only two possible outcomes (success or failure) and the probability of success is constant throughout the trials.

Question 18

The variable Employment Status,which can take either the value 1 for Employed and 0 for Unemployed,is an example of a _____________ random variable.
A)Poisson
B)Discrete
C)Hypergeometric
D)Continuous

Accepted Answer

The variable Employment Status can only take two possible values, Employed (1) or Unemployed (0), thus it is a discrete random variable.

Question 19

The distribution whose mean is equal to its variance is the _________ distribution.
A)Binomial
B)Poisson
C)Hypergeometric
D)Continuous

Accepted Answer

The Poisson distribution has the property that its mean and variance are equal, which is often denoted as λ. That is, E(X) = Var(X) = λ. Therefore, option B is the best choice.

Question 20

A discrete variable that can often be used to describe the number of occurrences of an event over a specified interval of time or space is a ___________ random variable.
A)Poisson
B)Discrete
C)Hypergeometric
D)Continuous

Accepted Answer

A Poisson random variable is used to model the number of occurrences of an event over a specified interval of time or space. The Poisson distribution is a discrete probability distribution that describes the probability of a given number of events occurring in a fixed interval of time or space, given the mean rate of occurrence. Therefore, the best choice is A, Poisson.

Quiz 5: Discrete Random Variables

The Securities and Exchange Commission has determined that the number of companies listed on the NYSE declaring bankruptcy is approximately a Poisson distribution with a mean of 2.6 per month.Find the probability that more than 1 bankruptcy occurs next month.

The Securities and Exchange Commission has determined that the number of companies listed on the NYSE declaring bankruptcy is approximately a Poisson distribution with a mean of 2.6 per month.Find the probability that exactly 4 bankruptcies occur next month.

If n = 15 and p = .4,then the standard deviation of the binomial distribution is:

The requirement that the probability of success remains constant from trial to trial is a property of the _______________ distribution.

If x is a binomial random variable,then the standard deviation of x is given by:

A fair die is rolled 10 times.What is the average number of even number outcomes?

A fair die is rolled 36 times.What is the standard deviation of the even number (2,4,or 6) outcomes?

The binomial distribution is characterized by situations that are analogous to

The equation for the variance of the binomial distribution is given by:

The Securities and Exchange Commission has determined that the number of companies listed on the NYSE declaring bankruptcy is approximately a Poisson distribution with a mean of 2.6 per month.Find the probability that no more than one bankruptcy occurs next month.

A fair die is rolled 10 times.What is the probability that an odd number (1,3,or 5) will occur fewer than 3 times?

When p = .5,the binomial distribution will _________ be symmetric.

Two characteristics,or assumptions,of the Poisson distribution are:

Which of the following statements about the binomial distribution is not correct?

A fair die is rolled 10 times.What is the probability that an even number (2,4,or 6) will occur between 2 and 4 times?

For a random variable X,the mean value of the squared deviations of its values from their expected value is called its ____________.

A random variable that is defined to be the total number of successes in n trials is a __________ random variable.

The variable Employment Status,which can take either the value 1 for Employed and 0 for Unemployed,is an example of a _____________ random variable.

The distribution whose mean is equal to its variance is the _________ distribution.

A discrete variable that can often be used to describe the number of occurrences of an event over a specified interval of time or space is a ___________ random variable.