Exam 7: Random Variables and Discrete Probability Distributions

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 standard deviation of X ?

The Binomial distribution and the Poisson distribution are discrete distributions.

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?

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.

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 X Y. b. Check whether E(XY)=E(X) ' E(Y) by separately evaluating each side of the equality.

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 ?

Gender is an example of a continuous random variable.

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 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: $\text { Hint: } \operatorname { corr } ( X , Y ) = \operatorname { Cov } ( X , Y ) / ( ( \sqrt { V ( X ) } \sqrt { V ( Y ) } ) = ( E ( X Y ) - E ( X ) E ( Y ) ) / ( \sqrt { V ( X ) } \sqrt { V ( Y ) } )$

An advertising executive receives an average of 10 telephone calls each afternoon between 2 and 4pm. The calls occur randomly and independently of one another. a. Find the probability that the executive will receive 13 calls between 2 and 4pm on a particular afternoon. b. Find the probability that the executive will receive seven calls between 2 and 3pm on a particular afternoon. c. Find the probability that the executive will receive at least five calls between 2 and 4pm on a particular afternoon.

The number of accidents that occur annually on a busy stretch of highway is an example of:

Which of the following best describes a function that assigns a numerical value to each simple event in a sample space?

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) 0.25 0.33 0.17 0.15 0.10 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?

Which of the following is not a characteristic of a binomial experiment?

An official from the Australian Securities and Investments Commission estimates that 75% of all investment bankers have profited from the use of insider information. If 15 investment bankers are selected at random from the Commission's registry, find the probability that: a. at most 10 have profited from insider information. b. at least six have profited from insider information. c. all 15 have profited from insider information.

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?

Given a binomial random variable with n = 20 and p = 0.6, find the following probabilities using the binomial table. a. P(X  13). b. P(X  15). c. P(X = 17). d. $P ( 11 < X < 14 )$

A binomial distribution for which the number of trials n is large can well be approximated by a Poisson distribution when the probability of success, p, is:

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 $P ( Y = 2 \mid X = 1 )$ d. Find the probability distribution of the random variable X+Y. e. Find E(XY). f. Find COV(X, Y).

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.