An analysis of the stock market produces the following information about the returns of two stocks: $\begin{array}{ll} &\text { Stock } 1 \text { Stock } 2\\ \begin{array}{ll} \text {Expected Returns }\\\text { Standard Deviations} \end{array}&\begin{array}{|cc|} \hline 15 \% & 18 \% \\ 20 & 32 \\ \hline \end{array} \end{array}$ 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.

a. The expected return on the portfolio is E(0.5X 1 + 0.5X 2 ) = 0.5E(X 1 ) + 0.5E(X 2 )= 16.5%. The variance of the portfolio's return is (0.5 * 20) 2 + (0.5 * 32) 2 + 2(0.5) 2 * 0.80 * 20 * 32 = 612. The standard deviation of the portfolio's return when $\rho$ 12 = 0.80 is therefore 24.74%. b. Your choice of investment in stock 1, the portfolio, or stock 2 depends on your desired level of risk (variance of return). The higher the risk you choose, the higher will be the expected return.

Which of the following best describes a discrete random variable? $\begin{array}{|l|l|} \hline A&\text {A random variable that can only assume a countable number of values. }\\ \hline B&\text {A random variable that can only assume an uncountable number of values. }\\ \hline C&\text { All random variable are discrete random variables}\\ \hline D&\text { None of these choices are correct}\\ \hline \end{array}$

The answer of Which of the following best describes a...

Which of the following best describes a function that assigns a numerical value to each simple event in a sample space? $\begin{array}{|l|l|} \hline A&\text {An expected value.}\\ \hline B&\text {The mean. }\\ \hline C&\text { A random variable. }\\ \hline D&\text {The standard deviation. }\\ \hline \end{array}$

The answer of Which of the following best describes a...

Which probability distribution is appropriate when the events of interest occur randomly, independently of one another, and rarely? $\begin{array}{|l|l|} \hline A&\text {Binomial distribution. }\\ \hline B&\text { Poisson distribution.}\\ \hline C&\text {Any discrete distribution. }\\ \hline D&\text { Any continuous distribution.}\\ \hline \end{array}$

The answer of Which probability distribution is appropriate when the...

The expected value, E(X), of a binomial probability distribution is: $\begin{array}{|l|l|} \hline A&n+p \\ \hline B&n p(1-p)\\ \hline C&n p\\ \hline D&n+p-1\\ \hline \end{array}$

The answer of The expected value, E(X), of a binomial...

The probability distribution for X is as follows: \[\begin{array} { l | r r r r } x & - 1 & 0 & 1 & 2 \\ \hline p ( x ) & 0.1 & 0.25 & 0.55 & 0.1 \end{array}\] Find the expected value of Y = X + 10.

The answer of The probability distribution for X is as...

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 $\sigma$.

a. E[X] = 2.57 in hundreds of newspapers

The joint probability distribution of X and Y is shown in the following table. \[\begin{array} { c | c c c } & & X \\ \hline Y&1&2 & 3 \\ \hline 1 & .30 & .18 & .12 \\ 2 & .15 & .09 & .06 \\ 3 & .05 & .03 & .02 \end{array}\] a. Determine the marginal probability distributions of X and Y. b. Are X and Y independent? Explain. c. Find P(Y = 2 | X = 1). d. Find the probability distribution of the random variable X + Y. e. Find E(XY). f. Find COV(X, Y).

a. @#IMG-DLM& b. Yes, because p(x, y) =

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: $\begin{array}{|l|l|} \hline A&\text { variance.}\\ \hline B&\text {standard deviatio: }\\ \hline C&\text {expected value. }\\ \hline D&\text { covariance.}\\ \hline \end{array}$

The answer of The weighted average of the possible values...

A Poisson distribution with $\mu$ = .60 is a: $\begin{array}{|l|l|} \hline A&\text { symmetrical distribution.}\\ \hline B&\text {negatively skewed distribution (skewed to the left). }\\ \hline C&\text {positively skewed distribution (skewed to the right). }\\ \hline D&\text {binomial distribution. }\\ \hline \end{array}$

The answer of A Poisson distribution with $\mu$ = .60...

Exam 7: Random Variables and Discrete Probability Distributions

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?

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Question 1

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a. μ= 5(30/60) = 2.5; P(X = 2) = 0.2565.
b. μ = 15(30/60) = 7.5; P(X = 8) = 0.1373.
c. μ = 4(30/60) = 2.0; P(X = 0) = 0.1353.

An analysis of the stock market produces the following information about the returns of two stocks: Stock 1 Stock 2 Expected Returns Standard Deviations 15\% 18\% 20 32 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.

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Question 2

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a. The expected return on the portfolio is E(0.5X₁ + 0.5X₂) = 0.5E(X₁) + 0.5E(X₂)= 16.5%.
The variance of the portfolio's return is (0.5 * 20)² + (0.5 * 32)² + 2(0.5)² * 0.80 * 20 * 32 = 612.
The standard deviation of the portfolio's return when $\rho$ ₁₂ = 0.80 is therefore 24.74%.
b. Your choice of investment in stock 1, the portfolio, or stock 2 depends on your desired level of risk (variance of return). The higher the risk you choose, the higher will be the expected return.

Let X represent the number of computers in Australian households who own computers. The probability distribution of X is as follows: x 1 2 3 4 5 p(x) .25 .33 .17 .15 .10 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?

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Question 3

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a. 0.42.
b. 0.75.
c. 0.58

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

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Question 4

The Binomial distribution and the Poisson distribution are discrete bivariate distributions.

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Question 5

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.

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Question 6

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.

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Question 7

The expected value, E(X), of a binomial probability distribution is: A n+p B np(1-p) C np D n+p-1

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Question 8

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).

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Question 9

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

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Question 10

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 : X P[X] 0 0.95 10 0.049 100 0.001 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 tun, the average prize won per \ 1 played is \ 0.59 . D None of these choices are correct

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Question 11

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?

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Question 12

The probability distribution for X is as follows: x -1 0 1 2 p(x) 0.1 0.25 0.55 0.1 Find the expected value of Y = X + 10.

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Question 13

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

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Question 14

The joint probability distribution of X and Y is shown in the following table. X Y 1 2 3 1 .30 .18 .12 2 .15 .09 .06 3 .05 .03 .02 a. Determine the marginal probability distributions of X and Y. b. Are X and Y independent? Explain. c. Find P(Y = 2 | X = 1). d. Find the probability distribution of the random variable X + Y. e. Find E(XY). f. Find COV(X, Y).

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Question 15

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.

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Question 16

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).

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Question 17

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 $\ge$ 5) d. P(6 $\le$ X $\le$ 10)

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Question 18

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 deviatio: C expected value. D covariance.

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Question 19

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.

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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

The Binomial distribution and the Poisson distribution are discrete bivariate distributions.

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.

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.

The expected value, E(X), of a binomial probability distribution is: A n+p B np(1-p) C np D n+p-1

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).

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

The probability distribution for X is as follows: x -1 0 1 2 p(x) 0.1 0.25 0.55 0.1 Find the expected value of Y = X + 10.

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

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.

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).

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 $\ge$ 5) d. P(6 $\le$ X $\le$ 10)

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 deviatio: C expected value. D covariance.

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.

What Is Statistics

Types of Data, Data Collection and Sampling

Graphical Descriptive Methods Nominal Data

Graphical Descriptive Techniques Numerical Data

Numerical Descriptive Measures

Probability

Continuous Probability Distributions

Statistical Inference: Introduction

Sampling Distributions

Estimation: Describing a Single Population

Estimation: Comparing Two Populations

Hypothesis Testing: Describing a Single Population

Hypothesis Testing: Comparing Two Populations

Inference About Population Variances

Analysis of Variance

Additional Tests for Nominal Data: Chi-Squared Tests

Simple Linear Regression and Correlation

Multiple Regression

Model Building

Nonparametric Techniques

Statistical Inference: Conclusion

Time-Series Analysis and Forecasting

Index Numbers

Decision Analysis

Filters

Exam 7: Random Variables and Discrete Probability Distributions

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

The Binomial distribution and the Poisson distribution are discrete bivariate distributions.

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.

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.

The expected value, E(X), of a binomial probability distribution is: A n+p B np(1-p) C np D n+p-1

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).

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

The probability distribution for X is as follows: x -1 0 1 2 p(x) 0.1 0.25 0.55 0.1 Find the expected value of Y = X + 10.

The probability distribution for X ,daily demand of a particular newspaper at a local newsagency,( in hundreds) is as follows: x 1 2 3 4 p(x) 0.05 0.42 0.44 0.09 a. Find and interpret the expected value of X. b. Find V(X). c. Find σ\sigmaσ .

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.

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).

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 ≥\ge≥ 5) d. P(6 ≤\le≤ X ≤\le≤ 10)

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 deviatio: C expected value. D covariance.

A Poisson distribution with μ\muμ = .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.

What Is Statistics

Types of Data, Data Collection and Sampling

Graphical Descriptive Methods Nominal Data

Graphical Descriptive Techniques Numerical Data

Numerical Descriptive Measures

Probability

Continuous Probability Distributions

Statistical Inference: Introduction

Sampling Distributions

Estimation: Describing a Single Population

Estimation: Comparing Two Populations

Hypothesis Testing: Describing a Single Population

Hypothesis Testing: Comparing Two Populations

Inference About Population Variances

Analysis of Variance

Additional Tests for Nominal Data: Chi-Squared Tests

Simple Linear Regression and Correlation

Multiple Regression

Model Building

Nonparametric Techniques

Statistical Inference: Conclusion

Time-Series Analysis and Forecasting

Index Numbers

Decision Analysis

Filters

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).

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

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).

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 $\ge$ 5) d. P(6 $\le$ X $\le$ 10)

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.