Question 1

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING:
Suppose X and Y are two random variables with E(X)= 1.50,E(Y)= 0.55,E(XY)= 0.80,Var(X)= 0.25,and Var(Y)= 0.2475.
-What is the value of Var(2X - 3Y)?

Accepted Answer

The answer of THE NEXT QUESTIONS ARE BASED ON THE...

Question 2

The covariance of two variables X and Y:

Accepted Answer

The covariance can be positive or negative and its magnitude is not limited to a maximum of 1. It measures how two variables change together, and can take on any numerical value.

Question 3

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING:
Suppose X and Y are two random variables with E(X)= 1.50,E(Y)= 0.55,E(XY)= 0.80,Var(X)= 0.25,and Var(Y)= 0.2475.
-What is the value of E(2X - 3Y)?

Accepted Answer

E(2X - 3Y) = 2E(X) - 3E(Y) = 2(1.50) - 3(0.55) = 3 - 1.65 = 1.35.

Question 4

A pair of jointly distributed random variables are said to be independent if and only if their joint probability distribution is the product of their:

Accepted Answer

Independent random variables have marginal probability distributions that are not affected by the values of the other random variable. Therefore, their joint probability distribution can be expressed as the product of their marginal probability distributions.

Question 5

If X and Y are random variables with Var(X)= 7.5,Var(Y)= 6 and Cov(X,Y)= 4,then Var(2X + 3Y)is:

Accepted Answer

The variance of a linear combination of two random variables $X$ and $Y$, $Var(aX + bY)$, is given by $a^2Var(X) + b^2Var(Y) + 2abCov(X,Y)$. Substituting the given values, we get $Var(2X + 3Y) = 2^2 \times 7.5 + 3^2 \times 6 + 2 \times 2 \times 3 \times 4 = 30 + 54 + 48 = 132$.

Question 6

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING:
Suppose X and Y are two random variables with E(X)= 1.50,E(Y)= 0.55,E(XY)= 0.80,Var(X)= 0.25,and Var(Y)= 0.2475.
-What is the value of Corr(X,Y)?

Accepted Answer

The correlation coefficient, Corr(X,Y), is calculated using the formula: Corr(X,Y) = [E(XY) - E(X)E(Y)] / sqrt(Var(X)Var(Y)). Substituting the given values, we get Corr(X,Y) = [0.80 - (1.50*0.55)] / sqrt(0.25*0.2475) = -0.1005.

Question 7

Given a Poisson random variable X,where the average number of times an event occurs in a certain period of time is 2.5,then P(X = 0)is:

Accepted Answer

The probability mass function for a Poisson random variable is given by P(X=k) = (e^-λ * λ^k) / k!, where λ is the expected value of X. Here, λ=2.5 and we want to find P(X=0). 
P(X=0) = (e^-2.5 * 2.5^0) / 0! = e^-2.5 ≈ 0.0821 (using a calculator or the approximation e^-x ≈ 1-x when x is small). Therefore, the best choice is (B).

Question 8

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING:
Suppose X and Y are two random variables with E(X)= 1.50,E(Y)= 0.55,E(XY)= 0.80,Var(X)= 0.25,and Var(Y)= 0.2475.
-What is the value of Cov(X,Y)?

Accepted Answer

Cov(X,Y) = E(XY) - E(X)E(Y) = 0.80 - (1.50 * 0.55) = 0.80 - 0.825 = -0.025.

Question 9

For two independent random variables X and Y,which of the following identities is always true?

Accepted Answer

DC) The variance of a linear combination of independent random variables is the sum of their variances each multiplied by the square of their coefficients. Thus, $Var(2X + 3Y) = 4Var(X) + 9Var(Y)$.D) The expected value of a linear combination of random variables is the linear combination of their expected values. Therefore, $E(2X + 3Y) = 2E(X) + 3E(Y)$, which can also be expressed as $5E(X) + 5E(Y)$ if $E(X) = E(Y)$, making the statement potentially misleading without this clarification. However, the correct interpretation based on the linearity of expectation is $E(2X + 3Y) = 2E(X) + 3E(Y)$, not directly $5E(X+Y)$, indicating a mistake in my initial assessment. The correct interpretation should focus on the linearity principle directly without implying equal expectations for $X$ and $Y$.

Question 10

The measure of linear association between two random variables that represents their joint variability is called:

Accepted Answer

Covariance measures the extent to which two random variables change in tandem from their expected values. It quantifies the degree to which two variables are related, with a positive covariance indicating a positive relationship and a negative covariance indicating a negative relationship. Choice A is incorrect as hypergeometric probability is a distribution that gives the probability of k successes in n draws from a finite population without replacement. Choice B is incorrect as conditional probability is the probability of an event occurring given that some other event has occurred. Choice D is incorrect as mean is a measure of central tendency and does not represent the joint variability between two random variables.

Question 11

If X and Y are any random variables with Cov(X,Y)= 0.25,  = 0.36,and 
= 0)49,then the coefficient of correlation p is:

Accepted Answer

The formula for the coefficient of correlation is p = Cov(X,Y)/[σ(X)σ(Y)]. We are given Cov(X,Y) = 0.25, so we need to find σ(X) and σ(Y). To do this, we use the formulas Var(X) = E(X^2) - [E(X)]^2 and Var(Y) = E(Y^2) - [E(Y)]^2. We are given E(X) = E(Y) = 0, and we can find E(X^2) and E(Y^2) by using the formulas Var(X) + [E(X)]^2 = E(X^2) and Var(Y) + [E(Y)]^2 = E(Y^2). Substituting the given values, we get Var(X) = 0.36 - 0^2 = 0.36 and Var(Y) = 0.49 - 0^2 = 0.49. Thus, σ(X) = sqrt(Var(X)) = 0.6 and σ(Y) = sqrt(Var(Y)) = 0.7. Substituting these values into the formula for p, we get p = 0.25/(0.6*0.7) = 0.595. Therefore, the answer is D.

Question 12

If X and Y are random variables with E(X)= 5 and E(Y)= 8,then E(2X + 3Y)is:

Accepted Answer

The expectation operator E is linear, so E(2X + 3Y) = 2E(X) + 3E(Y) = 2*5 + 3*8 = 10 + 24 = 34.

Question 13

Computation of marginal probabilities,mean,and variances for jointly distributed random variables can be developed in:

Accepted Answer

Marginal probabilities, mean, and variances for joint distributions can be quite complex and time-consuming to calculate by hand. Therefore, the use of various computer packages is the best choice. These computer packages have easy-to-use functions and algorithms that can quickly and accurately compute the desired values.

Question 14

A random variable is a variable that takes on numerical values realized by the outcomes in the sample space generated by a random experiment.

Accepted Answer

This is the correct definition of a random variable.

Question 15

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING:
Suppose X and Y are two random variables with E(X)= 1.50,E(Y)= 0.55,E(XY)= 0.80,Var(X)= 0.25,and Var(Y)= 0.2475.
-What can you say about the relationship between X and Y?

Accepted Answer

The answer of THE NEXT QUESTIONS ARE BASED ON THE...

Question 16

If X and Y are random variables,the sum of all the conditional probabilities of X given a specific value of Y will always be:

Accepted Answer

The sum of all conditional probabilities of X given a specific value of Y is equal to the marginal probability of X, which sums to 1 by the definition of probability. Therefore, the correct answer is C.

Question 17

The collections of financial instruments that each have returns defined by a probability distribution and used to obtain a combined investment with expected risks and returns are called:

Accepted Answer

A portfolio is a collection of financial instruments such as stocks, bonds, and cash equivalents, with the objective of combining their risks and returns. It is constructed by an investor or a fund manager in a way that optimizes the expected return based on the investor's risk tolerance. Therefore, portfolios are the best choice for the described scenario. The other options, folders, archives, and probability investments, do not have a specific financial meaning related to the situation described.

Question 18

If X and Y are independent random variables,which of the following identities is false?

Accepted Answer

If X and Y are independent, then Cov(X,Y) = 0. Therefore, option A is false. Option B is true as independent random variables have zero covariance. Options C and D are true for all random variables, regardless of their independence.

Question 19

In dealing with a pair of jointly distributed discrete random variables,the conditional probability distribution on one random variable,given specified values of another,is the:

Accepted Answer

The conditional probability distribution of one random variable given specified values of another random variable is a collection of conditional probabilities. It gives the probabilities of the possible values of one random variable given that the other random variable takes one of the specified values. It is calculated by dividing the joint probability distribution of the two random variables by the marginal probability distribution of the specified value of the other random variable. This collection of conditional probabilities forms a new probability distribution for the first random variable, conditioned on the value of the second random variable.

Question 20

The mean of a Poisson random variable X,where λ is the average number of times that an event occurs in a certain period of time or space,is:

Accepted Answer

The mean of a Poisson random variable X is λ, which is the average number of times that an event occurs in a certain period of time or space. This is a well-known property of the Poisson distribution.

Quiz 4: Discrete Random Variables and Probability Distributions

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING: Suppose X and Y are two random variables with E(X) = 1.50,E(Y) = 0.55,E(XY) = 0.80,Var(X) = 0.25,and Var(Y) = 0.2475. -What is the value of Var(2X - 3Y) ?

The covariance of two variables X and Y:

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING: Suppose X and Y are two random variables with E(X) = 1.50,E(Y) = 0.55,E(XY) = 0.80,Var(X) = 0.25,and Var(Y) = 0.2475. -What is the value of E(2X - 3Y) ?

A pair of jointly distributed random variables are said to be independent if and only if their joint probability distribution is the product of their:

If X and Y are random variables with Var(X) = 7.5,Var(Y) = 6 and Cov(X,Y) = 4,then Var(2X + 3Y) is:

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING: Suppose X and Y are two random variables with E(X) = 1.50,E(Y) = 0.55,E(XY) = 0.80,Var(X) = 0.25,and Var(Y) = 0.2475. -What is the value of Corr(X,Y) ?

Given a Poisson random variable X,where the average number of times an event occurs in a certain period of time is 2.5,then P(X = 0) is:

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING: Suppose X and Y are two random variables with E(X) = 1.50,E(Y) = 0.55,E(XY) = 0.80,Var(X) = 0.25,and Var(Y) = 0.2475. -What is the value of Cov(X,Y) ?