Exam 7: A: Random Variables and Discrete Probability Distributions

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} Calculate E(XY).

Returns on Investment An analysis of the stock market produces the following information about the returns of two stocks. Stock 1 Stok 2 Bxpseted Returns 15\% 18\% Standand Deviations 20 32 Assume that the returns are positively correlated with correlation coefficient of 0.80. -{Returns on Investment Narrative} Find the standard deviation of the return on a portfolio consisting of an equal investment in each of the two stocks.

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} Compute the mean and variance of X.

Number of Hamsters The joint probability distribution of X and Y is shown in the accompanying table,where X denotes the number of hamsters that Quinn may have next year,and Y denotes the number of hamsters that her boyfriend,Jason,may have when she moves in with him next year. $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ X 1 1 0.4 0.1 2 0.3 0.2 -{Number of Hamsters Narrative} Calculate E(XY).

Blackjack The probability distribution of a random variable X is shown below,where X represents the amount of money (in $1,000s)gained or lost in a particular game of Blackjack. -4 0 4 8 p(x) 0.15 0.25 0.20 0.40 -{Blackjack Narrative} Find the following probabilities: a. P(X $\le$ 0) b. P(X > 3) c. P(0 $\le$ X $\le$ 4) d. P(X = 5)

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 ____________________ of X and Y is a measure of the strength and direction of the linear relationship between X and Y.It is easy to put into perspective because it is always between -1 and 1.

A portfolio return,R_p,of two stocks with individual returns,R₁ and R₂,is,in general,given by R_p = R₁ + R₂.

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} Calculate E(X + Y)directly by using the probability distribution of X + Y.

In the notation below,X is the random variable,E and V refer to the expected value and variance,respectively.Which of the following is false?

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 two variables with $\sigma _ { x } ^ { 2 } = 12.25$ , $\sigma _ { y } ^ { 2 } = 17.64$ ,and COV(X,Y)= 11.76,then the coefficient of correlation $\sigma$ = 0.8.

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 V(X + Y)directly by using the probability distribution of X + Y.

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 you add two single probability distributions together you get a bivariate distribution.

The dean of students conducted a survey on campus.Grade point average (GPA)is an example of a(n)____________________ random variable.

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} Calculate V(X + Y)directly by using the probability distribution of X + Y.

Katie's Portfolio Katie is given the following information about the returns on two stocks: E(R₁)= 0.10,E(R₂)= 0.15,V(R₁)= 0.0225,and V(R₂)= 0.0441. -{Katie's Portfolio Narrative} Compute the expected value of the portfolio composed of 30% stock 1 and 70% stock 2.

Which of the following is a discrete random variable?

If n = 10 and p = 0.60,then the mean of the binomial distribution is

Online Bankers An official from the securities commission estimates that 75% of all online bankers have profited from the use of insider information.Assume that 15 online bankers are selected at random from the commission's registry. -{Online Bankers Narrative} Find the probability that at most 10 have profited from insider information.

Number of Hamsters The joint probability distribution of X and Y is shown in the accompanying table,where X denotes the number of hamsters that Quinn may have next year,and Y denotes the number of hamsters that her boyfriend,Jason,may have when she moves in with him next year. $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ X 1 1 0.4 0.1 2 0.3 0.2 -{Number of Hamsters Narrative} Are X and Y independent? Explain.

Montana Highways A recent survey in Montana revealed that 60% of the vehicles traveling on highways,where speed limits are posted at 70 miles per hour,were exceeding the limit.Suppose you randomly record the speeds of ten vehicles traveling on US 131 where the speed limit is 70 miles per hour.Let X denote the number of vehicles that were exceeding the limit. -{Montana Highways Narrative} Find P(X = 10).