Exam 7: Random Variables and Discrete Probability Distributions

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

A table,formula,or graph that shows all possible values a random variable can assume,together with their associated probabilities,is referred to as probability distribution.

​ Post office The number of arrivals at a local post office between 3:00 and 5:00 P.M.has a Poisson distribution with a mean of 12. ​ ​ -{{Post Office Narrative} Find the probability that the number of arrivals between 4:00 and 5:00 P.M.is exactly two.

The binomial random variable is the number of successes that occur in a fixed period of time.

Returns on Investment An analysis of the stock market produces the following information about the returns of two stocks. 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.

One requirement of bivariate distributions is that 0 ≤ P(x,y)≤ 1 for all pairs of values (x,y).

The covariance can be negative but the coefficient of correlation cannot.

The Poisson probability distribution is a continuous probability distribution.

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. ​ ​ -{Golfing Store Narrative} Determine the marginal probability distributions of X and Y.

The number of homeless people in Boston is an example of a discrete random variable.

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

​ Post office The number of arrivals at a local post office between 3:00 and 5:00 P.M.has a Poisson distribution with a mean of 12. ​ ​ -{Post Office Narrative} Find the probability that the number of arrivals between 3:30 and 4:00 P.M.is at least 10.

Stress Consider a binomial random variable X with n = 5 and p = 0.40,where X represents the number of times in the final exam week a student with 18 credit hours may feel stressed. ​ ​ -{Stress Narrative} Find P(X < 3).

Number of Horses ​ The random variable X represents the number of horses per family in a rural area in Iowa,with the probability distribution: p(x)= 0.05x,x = 2,3,4,5,or 6. ​ ​ -{Number of Horses Narrative} Find the variance and standard deviation of X.

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.

A probability distribution for a single random variable is referred to as a(n)____________________ distribution.

If X and Y are two variables with ,and COV(X,Y)= 11.76,then the coefficient of correlation ρ = 0.8.

Stress Consider a binomial random variable X with n = 5 and p = 0.40,where X represents the number of times in the final exam week a student with 18 credit hours may feel stressed. ​ ​ -{Stress Narrative} Find the probability distribution of X.

Classified Department Phone Calls A classified department receives an average of 10 telephone calls each afternoon between 2 and 4 P.M.The calls occur randomly and independently of one another. ​ ​ -{Classified Department Phone Calls Narrative} Find the probability that the department will receive seven calls between 2 and 3 P.M.on a particular afternoon.

Number of Motorcycles The probability distribution of a discrete random variable X is shown below,where X represents the number of motorcycles owned by a family. ​ ​ -{Number of Motorcycles Narrative} Apply the laws of expected value to find the following: a.E(X²) b.E(2X² + 5) c.E(X − 2)²