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Exam 7: Random Variables and Discrete Probability Distributions

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Question 31

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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 $\sigma$ .

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

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Question 32

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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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a. P(X = 0) = 0.778, P(X = 1) ...

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Question 33

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If X and Y are two independent random variables with V(X) = 6 and V(Y) = 5, then V(3X + 2Y) is: $\begin{array} { | l | l | } \hline \text { A. } & 11 . \\\hline \text { B. } & 158 . \\\hline \text { C. } & 28 . \\\hline \text { D. } & 74 . \\\hline\end{array}$

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?

The expected number of heads in 90 tosses of an unbiased coin is: $\begin{array} { | l | l | } \hline \text { A. } & 30 . \\\hline \text { B. } & 45 . \\\hline \text { C. } & 50 . \\\hline \text { D. } & 60 . \\\hline\end{array}$

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

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.

State whether or not each of the following are valid probability distributions, and if not, explain why not. a. $\begin{array}{c|cccc}x & 0 & 1 & 2 & 3 \\\hline p(x) & .15 & .25 & .35 & .45\end{array}$ b. $\begin{array}{c|cccc}x & 2 & 3 & 4 & 5 \\\hline p(x) & -.10 & .40 & .50 & .25\end{array}$ c. $\begin{array}{c|ccccc}x & -2 & -1 & 0 & 1 & 2 \\\hline p(x) & .10 & .20 & .40 & .20 & .10\end{array}$

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.

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

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).

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.

The joint probability distribution of X and Y is shown in the following table. $\quad$ $\quad$ $\quad$ $\quad$$X$ $\begin{array}{c|ccc}Y & 1 & 2 & 3 \\\hline 1 & .30 & .18 & .12 \\2 & .15 & .09 & .06 \\3 & .05 & .03 & .02\end{array}$ 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 table, formula, or graph that shows all possible countable values a random variable can assume, together with their associated probabilities, is called a:

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: