Exam 7: Random Variables and Discrete Probability Distributions

The variance of the sum always equals the sum of the variances.

Elizabeth's Portfolio ​ Elizabeth has decided to form a portfolio by putting 30% of her money into stock 1 and 70% into stock 2.She assumes that the expected returns will be 10% and 18%,respectively,and that the standard deviations will be 15% and 24%,respectively. ​ ​ -{Elizabeth's Portfolio Narrative} Compute the standard deviation of the returns on the portfolio assuming that the two stocks' returns are perfectly positively correlated.

A lab at the DeBakey Institute orders 150 rats a week for each of the 52 weeks in the year for experiments that the lab conducts.Suppose the mean cost of rats used in lab experiments turned out to be $20.00 per week.Interpret this value.

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. ​ ​ -{Retries Narrative} What is the standard deviation of the number of retries?

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 minimizing her risk,which stock should she choose?

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} What is the expected number of Online bankers who have profited from the use of insider information?

A motorcycle insurance company evaluates many numerical variables about a person before deciding on an appropriate rate for motorcycle insurance.How long a person has been a licensed rider is an example of a(n)____________________ 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 variance and standard deviation.

For each of the following random variables,indicate whether the variable is discrete or continuous,and specify the possible values that it can assume. a.X = the number of traffic accidents in Albuquerque on a given day. b.X = the amount of weight lost in a month by a randomly selected dieter. c.X = the average number of children per family in a random sample of 175 families. d.X = the number of households out of 10 surveyed that own a convection oven. e.X = the time in minutes required to obtain service in a restaurant.

The Poisson distribution is applied to events for which the probability of occurrence over a given span of time,space,or distance is very small.

The number of customers making a purchase out of 30 randomly selected customers has a Poisson distribution.

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} Assume that X and Y are independent and find their bivariate distribution.

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 no more than 1 incident in a year.

Bivariate distributions provide probabilities of combinations of two variables.

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 all 15 have profited from insider information.

A continuous variable may take on any value within its relevant range even though the measurement device may not be precise enough to record it.

Another name for the mean of a probability distribution is its expected value.

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

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} Find the following probabilities: a.P(X > 1) b.P(X ≤ 2) c.P(1 ≤ X ≤ 2) d.P(0 < X < 1) e.P(1 ≤ X < 3)

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).