Exam 7: Random Variables and Discrete Probability Distributions

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

NARRBEGIN: Gym Visits 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 NARREND -{Gym Visits Narrative} What is the probability that the student visits the gym at least once in a month?

NARRBEGIN: Golfing Store 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. 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 NARREND -{Golfing Store Narrative} Calculate the expected values of X and Y.

NARRBEGIN: Number of Hamsters 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. X 1 1 0.4 0.1 2 0.3 0.2 NARREND -{Number of Hamsters Narrative} Compute the covariance and the coefficient of correlation.

NARRBEGIN: Shopping Outlet 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 \square.20 0.15 NARREND -{Shopping Outlet Narrative} Calculate the variance and standard deviation of Y directly from the probability distribution of Y.

NARRBEGIN: Number of Birds 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 NARREND -{Number of Birds Narrative} Verify that V(X + Y) = V(X) + V(Y). Did you expect this result? Why?

NARRBEGIN: 911 Phone Calls 911 Phone Calls 911 phone calls arrive at the rate of 30 per hour at the local call center.NARREND -{911 Phone Calls Narrative} Find the probability of receiving exactly eight calls in 15 minutes.

The probability P(X $\le$ x) is called a(n) ____________________ probability. The binomial table reports these probabilities.

If the probability of success p remains constant in a binomial distribution, an increase in n will not change the mean.

The Poisson random variable is a discrete random variable with infinitely many possible values.

NARRBEGIN: Risky Undertaking 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.NARREND -{Risky Undertaking Narrative} Find the expected value of the payoff from your investment of $2,000.

NARRBEGIN: Montana Highways 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.NARREND -{Montana Highways Narrative} Find P(3 $\le$ X $\le$ 6).

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

If X and Y are independent variables, then their coefficient of correlation $\rho$ = 0.

NARRBEGIN: Blackjack 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 NARREND -{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)

If X and Y are two variables with $\sigma _ { z } ^ { 2 } = 3.25$ , $\sigma _ { y } ^ { 2 } = 5.8$ , and COV(X, Y) = 14.703, then the coefficient of correlation $\rho$ = 0.78.

NARRBEGIN: Shopping Outlet 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 \square.20 0.15 NARREND -{Shopping Outlet Narrative} Find the variance and standard deviation of the number of stores entered.