Exam 7: Random Variables and Discrete Probability Distributions

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. ​ ​ -{Unsafe Levels of Radioactivity Narrative} Find the probability that there will be at least 1 incident in a year.

Let X be a Poisson random variable with μ = 8.Use the table of Poisson probabilities to calculate: a.P(X ≤ 6) b.P(X = 4) c.P(X ≥ 3) d.P(9 ≤ X ≤ 14)

Which of the following is a continuous random variable?

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 expected number of times a student may feel stressed during the final exam week.

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. ​ ​ -{Number of Birds Narrative} Compute the mean and variance of Y.

The number of home insurance policy holders is an example of a discrete random variable

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

If X has a binomial distribution with n = 4 and p = 0.3,find P(X > 1).

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(4 < X < 9).

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

Compute the following Poisson probabilities (to 4 decimal places)using the Poisson formula: a.P(X = 3),if μ = 2.5 b.P(X ≤ 1),if μ = 2.0 c.P(X ≥ 2),if μ = 3.0

If X and Y are any random variables with COV(X,Y)= 0.25, ,then the coefficient of correlation ρ is

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 expected number of horses per family.

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. ​ ​ -{Number of Birds Narrative} Verify that E(XY)= E(X)E(Y).Did you expect this result? Why?

Shopping Outlet A shopping outlet estimates the probability distribution of the number of stores shoppers actually enter as shown in the table below. ​ ​ -{Shopping Outlet Narrative} Use the laws of variance to calculate the variance and standard deviation of Y from the probability distribution of X.

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. ​ ​ -{Unsafe Levels of Radioactivity Narrative} Find the variance of the number of incidents in one year.

In a Poisson experiment,the number of successes that occur in any interval of time is ____________________ of the number of success that occur in any other interval.

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 variance of a portfolio of two investments will be equal to the sum of the variances of the two investments plus twice the covariance between the investments.

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