Exam 7: A: Random Variables and Discrete Probability Distributions

Number of Birds Alana and Eva are sisters.Let X denote the number of birds that Alana may have in the next two years,and let Y denote the number of birds Eva may have during the same period.The marginal probability distributions of X and Y are shown below. 0 1 2 ( ) 0.5 0.3 0.2 y 0 1 2 ( ) 0.4 0.5 0.1 -{Number of Birds Narrative} Calculate E(XY)directly by using the probability distribution of XY.

Retries The following table contains the probability distribution for X = the number of retries necessary to successfully transmit a 1024K data package through a double satellite media. 0 1 2 3 ( ) 0.35 0.35 0.25 0.05 -The variance of X must be non-negative;the variance of Y must be non-negative;hence the covariance of X and Y must be non-negative.

The expected value,E(X),of a binomial probability distribution with n trials and probability p of success is:

Shopping Outlet A shopping outlet estimates the probability distribution of the number of stores shoppers actually enter as shown in the table below. 0 1 2 3 4 ( ) 0.05 0.35 0.25 0.20 0.15 -{Shopping Outlet Narrative} Find the variance and standard deviation of the number of stores entered.

Retries The following table contains the probability distribution for X = the number of retries necessary to successfully transmit a 1024K data package through a double satellite media. 0 1 2 3 ( ) 0.35 0.35 0.25 0.05 -If X and Y are any random variables with COV(X,Y)= 0.25, $\sigma _ { x } ^ { 2 } = 0.36$ ,and $\sigma _ { y } ^ { 2 } = 0.49$ ,then the coefficient of correlation $\rho$ is

The trials in a binomial experiment are ____________________,meaning the outcome of one trial does not affect the outcomes of any other trials.

Retries The following table contains the probability distribution for X = the number of retries necessary to successfully transmit a 1024K data package through a double satellite media. 0 1 2 3 ( ) 0.35 0.35 0.25 0.05 -Bivariate distributions provide probabilities of combinations of two variables.

Number of Birds Alana and Eva are sisters.Let X denote the number of birds that Alana may have in the next two years,and let Y denote the number of birds Eva may have during the same period.The marginal probability distributions of X and Y are shown below. 0 1 2 ( ) 0.5 0.3 0.2 y 0 1 2 ( ) 0.4 0.5 0.1 -{Number of Birds Narrative} Determine the probability distribution of the random variable X + Y.

Mobile Phones Sales After analyzing sales data,the owner of a Mobile Phone store produced the following joint probability distribution of the number of iPhones (X)and Blackberries (Y)sold daily. $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ X 1 1 0.4 0.1 2 0.3 0.2 -{Mobile Phones Sales Narrative} Compute the expected number of iPhones sold daily.

The following information regarding a portfolio of two stocks are given: w₁ = .25,w₂ = .75,E(R₁)= .08,and E(R₂)= .15.Which of the following regarding the portfolio expected return,E(R_p),is correct?

Golfing Store The joint probability distribution of variables X and Y is shown in the table below,where X is the number of drivers and Y is the number of putters sold daily in a small golfing store. $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ X 1 2 3 1 0.30 0.18 0.12 2 0.15 0.09 0.06 3 0.05 0.03 0.02 -{Golfing Store Narrative} Find the probability distribution of the random variable X + Y.

The binomial distribution deals with consecutive trials,each of which has two possible outcomes.

Car Sales The joint probability distribution of variables X and Y is shown in the table below.Rebecca and Rachel are car salespeople.Let X denote the number of cars that Rebecca will sell in a month,and let Y denote the number of cars Rachel will sell in a month. $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ X 1 2 3 1 0.30 0.18 0.12 2 0.15 0.09 0.06 3 0.05 0.03 0.02 -{Car Sales Narrative} Verify that V(X + Y)= V(X)+ V(Y).Did you expect this result? Why?

For a random variable X,V(X + 3)= V(X + 6),where V refers to the variance.

Sports Fans Suppose that past history shows that 5% of college students are sports fans.A sample of 10 students is to be selected. -{Sports Fans Narrative} Find the probability that more than 1 student is a sports fan.

If the covariance between two investments of a portfolio is zero,the variance of the portfolio will be equal to the sum of the variances of the investments.

A motorcycle insurance company evaluates many numerical variables about a person before deciding on an appropriate rate for motorcycle insurance.The number of tickets a person has received in the last 3 years is an example of a(n)____________________ random variable.

Unsafe Levels of Radioactivity The number of incidents at a nuclear power plant has a Poisson distribution with a mean of 6 incidents per year. -The time required to drive from New York to New Mexico is a discrete random variable.

Risky Undertaking Suppose you make a $2,000 investment in a risky undertaking.There is a 50% chance that the payoff from the investment will be $5,000,a 20% chance that you will just get your money back,and a 30% chance that you will receive nothing at all from your investment. -{Risky Undertaking Narrative} Find the expected value of the payoff from your investment of $2,000.

Gym Visits Let X represent the number of times a student visits a gym in a one month period.Assume that the probability distribution of X is as follows: 0 1 2 3 ( ) 0.05 0.25 0.50 0.20 -{Gym Visits Narrative} What is the probability that the student visits the gym at most twice in a month?