University of California Discrete Distributions & Bernoulli and Binomial Lab

1.A restaurant regards a patron as a loyal customer if they have spent more than $50 over the course of

a six months. Let X be the total amount, in dollars, a loyal customer spends at this restaurant in a six

month period. Let f(x) = c5/x^3and SX = [50, ?)

(a) For what value of c is f(x) a valid density?

(b) What is the expected amount spent, in dollars, of a loyal customer?

(c) What is the probability that a loyal customer spends exactly $50 over the course of six months?

(d) What is the probability that a loyal customer spends more than $ 100 and less than $ 200 in a year?

2. The total daily sugar intake of a young adult (20-34 year old) in Peru in 2015 was estimated to be

µ = 460 grams on average, with a standard deviation of ? = 109 grams. Suppose we are told that the

Normal distribution is a good approximation for this variable.

(a) What is the probability that a young Peruvian adult had a daily sugar intake between 250 and

400 grams? Write out the solution in integral form and use the following R code to compute the

numerical answer:pnorm(x2, mean = µ, sd = ?)=pnorm(x1, mean = µ, sd = ?)

(b) What daily sugar intake would you need to have so that the probability of a young Peruvian adult

to have had an intake less than yours is .45? Write out the equation we would need to solve in order

to answer this question, then use the qnorm() function in R to fifind this score.

(c) What is the probability that a young Peruvian adult had a daily sugar intake greater than 500

(d) What is the probability that a young Peruvian adult had a daily sugar intake less than 250 grams?

grams?

3. Let X be a continuous random variable. Let f(x) = cx2 and SX = [0, 3]

(a) What value of c will make f(x) a valid density?

(b) What is P(X = 1.75)?

(c) Find E(X).

(d) What is P(0.5 < X < 1)?

4. The time between a lighting strike in a thunderstorm follows an Exponential distribution with parameter

? =1/30 seconds.

(a) What is the probability that the next lightning strike is more than 15 seconds?

(b) What is the expected time (in seconds) until the lightning strike?

(c) It has already been 10 seconds since the last lightning strike. What is the probability that the next

(d) Suppose you wanted to watch the thunderstorm from a nearby window. How long do you expect it

5. It always takes some time to commute to campus. Depending on traffiffiffic and weather, the commute can

take anywhere from 15 to 25 minutes, with any time in that interval being equally likely.

(a) What is the probability that your commute is less than 8 minutes.

will take for you to observe two lightning strikes? How much do we expect this time to vary around

the expected time? Assume all lightning strikes are independent of one another.

lightning strike will not be until at least 26 seconds later?

(b) What is the expected amount of minutes it will take to commute to campus? Interpret this in the

context of the problem in a sentence or two.

(c) What is the probability that you will need between 12 and 22 minutes?

You have 8:30am classes three times a week and 8am classes two times a week, with each class being

independent from the other.

(d) You will be late to your 8:30am class if it takes more than 23 minutes to commute to campus. What

is the probability that you will be late to each of the 8:30am classes next week?

(e) You will be late to your 8am class if you need more than 17 and a half minutes to get to campus.

What is the probability that you are late to at least one 8am class next week?

6. The height of American men is modeled with a normal distribution in feet, with mean µ = 5.75 and

variance ?2 = 0.333.

(b) What is the probability that a man will be more than 6 feet and 9 inches tall?

(a) Write out the integral form of the probability that a man will be between 6 feet and 6 feet and 6

inches tall. Then use the pnorm function in R to fifind this probability

(c) What is the probability that a man is at most fifive foot seven?

(d) What is the maximum height needed for a man to be in the bottom 10% of men’s height?

7. The number of years before the next album release of your favorite band follows an exponential distri

bution. The band is expected to release an album every 2 years.

(a) What is the parameter of the Exponential distribution? What is it equal to in this problem?

(b) What is the probability that the next album is released in 4 to 5 years.

(c) It’s been 2 years since the last album was released. What is the probability that the next album

will be released in 6 or more years?