Exam 7: Random Variables and Discrete Probability Distributions

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.

A table, formula, or graph that shows all possible countable values a random variable can assume, together with their associated probabilities, is called a: A discrete probability distribution. B continuous probability distribution. C bivariate probability distribution. D probability tree.

State whether or not each of the following are valid probability distributions, and if not, explain why not. a. x 0 1 2 3 p(x) .15 .25 .35 .45 b. x 2 3 4 5 p(x) -.10 .40 .50 .25 C. x -2 -1 0 1 2 p(x) .10 .20 .40 .20 .10

Let X and Y be two independent random variables with the following probability distributions: x 1 2 3 p(x) 0.2 0.5 0.3 y -1 0 1 p(y) 0.3 0.3 0.4 a. Find the probability distribution of the random variable XY. b. Check whether E(XY) = E(X) ´ E(Y) by separately evaluating each side of the equality.

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).

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.

If X and Y are two independent random variables with V(X) = 6 and V(Y) = 5, then V(3X + 2Y) is: A. 11. B. 158. C. 28. D. 74

If X and Y are any random variables with E(X)= 3, E(Y) = 2, E(XY) = 12, V(X) = 16 and V(Y) = 25, then the relationship between X and Y is a: Hint: corr(X,Y) =( E(xy) - E(x)E(y))/ $\sqrt { V ( x ) V ( y ) }$ \begin{array}{|l|l|}\hline A&\text { weak negative relationship. }\\\hline B&\text {strong positive relationship. }\\\hline C&\text { strong negative relati on shi \mathrm{p} .}\\\hline D&\text { weak negative relati onship.}\\\hline \end{array}

The standard deviation of a binomial distribution for which n = 100 and p = .35 is: A. 4.77 B. 2.275. C. 47.7. D. 22.75.

For each of the following random variables, indicate whether the variable is discrete or continuous, and specify the possible values that it can assume. a. X = The number of animals visited by a veterinarian in one day. b. X = Closing share price of a particular stock over one month. c. X = The weights of new members at a gymnasium. d. X = The number of students attending a lecture, where the theatre seats 250 people. e. X = The temperature at a seaside resort in the summer.

Which of the following cannot generate a Poisson distribution? A The number of children watching a movie. B The number of telephone calls received by a switchboard in a specified time period. C The number of customers arriving at a petrol station on Christmas day. D The number of bacteria found in a cubic yard of soil.

Let X and Y be two independent random variables with the following probability distributions: x 1 2 3 p(x) 0.2 0.5 0.3 y -1 0 1 p(y) 0.3 0.3 0.4 Find the probability distribution of the random variable X + Y.

Let X represent the number of computers in an Australian household, for those that own a computer. x 1 2 3 4 5 p(x) .25 .33 .17 .15 .10 a. Find and interpret the expected number of computers in a randomly selected Australian household. b. Find the variance of the number of computers in a randomly selected Australian household.

If X and Y are random variables with E(X) =7 and E(Y) = 3, then E(2X + 3Y) is: D. 21. A. 10. B. 23. C. 27. D. 21.

A bivariate distribution is a distribution is a joint probability distribution of two variables.

The number of people winning a lottery ticket each week is an example of a Poisson variable.

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?

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?

The probability distribution for X is as follows: x -1 0 1 2 p(x) 0.1 0.25 0.55 0.1 a. Find E[5X + 1]. b. Find V[5X + 1].

The Poisson random variable is a: A discrete random variable with infinitely many possible values. B discrete random variable with a finite number of possible values. C continuous random variable with infinitely many possible values. D continuous random variable with a finite number of possible values.