Exam 7: A: Random Variables and Discrete Probability Distributions

The covariance between two investments of a portfolio is equal to the sum of the variances of the investments.

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} Determine the marginal probability distributions of X and Y.

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} Determine the marginal probability distribution of X.

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:

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 -The monthly sales at a Gas Station have a mean of $50,000 and a standard deviation of $6,000.Profits are calculated by multiplying sales by 40% and subtracting fixed costs of $12,000.Find the mean and standard deviation of monthly profits.

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} Verify that V(X + Y)= V(X)+ V(Y).Did you expect this result? Why?

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

The variance of a binomial distribution is equal to ____________________.

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. -Given that X is a discrete random variable,then the laws of expected value and variance can be applied to show that E(X + 5)= E(X)+ 5,and V(X + 5)= V(X)+ 25.

Which of the following are required conditions for the distribution of a discrete random variable X that can assume values x_i?

The following information regarding a portfolio of two stocks are given: w₁ = .65,w₂ = .35,E(R₁)= .12,and E(R₂)= .14.Which of the following regarding the portfolio expected return,E(R_p),is correct?

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 length of time for which an apartment in a large complex remains vacant is a discrete random variable.

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 = 3.8, $\sigma$ _y = 4.2,and COV(X,Y)= -0.25,then V(X + Y)= 31.58.

The expected number of heads in 250 tosses of an unbiased coin is 125.

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 = 2).

To find the probability that X is at most 10,you should find the probability that X is 10 or ____________________.

The probability of a failure in a binomial experiment is denoted by ____________________.

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} If Katie is most interested in maximizing her returns,which stock should she choose?

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} A sample of 100 students is to be selected.What is the average number that you would expect to sports fan?

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 at most 1 student is a sports fan.