Deck 7: Special Continuous Distribution

Question 1

Let X be a random number selected from the interval [-2,3] . Find the expected value of |X³| .

Answer 1

$\int _ { - 2 } ^ { 0 } \frac { - x ^ { 3 } } { 5 } d x + \int _ { 0 } ^ { 3 } \frac { x ^ { 3 } } { 5 } d x = 16 / 20 + 81 / 20 = 97 / 20$

Question 2

At a certain university, there are exactly 3000 male undergraduates, 3000 female undergraduates, and 3000 gender non-binary undergraduates. If a group of 50 undergraduates is formed at random, what is an approximation of the probability that at least 20 of the members are female?

Answer 2

P=0.1587

Question 3

For a certain model of oven, the temperature of the oven when set to is a uniform random variable over the interval (328,361). Find the proportion of such ovens that run hotter than the indicated temperature.

Answer 3

$\frac { 11 } { 33 }$ .

Question 4

For a customer, the time spent at a help desk with an associate is exponential with mean 120 minutes. If 5 customers are at the help desk being helped (independently) by five associates, find the probability that the waiting times of at least 2 of them are less that 120 minutes.

Question 5

Suppose that incomes in a certain area are normally distributed with mean $54,000 and standard deviation $3000 . Give the probability density function for the income of a randomly selected citizen in this region.

Question 6

Suppose that Y is an exponential random variable with parameterλ=1.5. Find P(Y>(E(X))- 1)

Question 7

Suppose that in a 45-mph zone, the speeds at which drivers are pulled over are normally distributed with mean 60 mph and variance 25 mph. What is the maximum speed a driver can go with at most a 10% probability of being pulled over?

Question 8

Let X be a uniform random variable over (0,1) . Find the expected value of the random variable $\frac { 1 } { 1 + X }$ .

Question 9

Every week, a station plays 310 blues songs and 620 jazz songs. If one day, a random block of 42 songs is played, what is the probability that at least 19 jazz and at least 19 blues songs are played?

Question 10

At a concert, the fraction of attendees who dance is a beta random variable X with parameters (4,3). Find the probability that out of 1000 concert attendees, at least 750 dance.

Question 11

When ordering pizza from a certain pizza parlor, the delivery time is an exponential random variable with mean 25 minutes. If it takes more than 30 minutes for the pizza to be delivered, they provide it for free. If it takes more than 40 minutes, not only will the customer get the pizza for free, but he or she will receive an additional $10. If a pizza with delivery costs $18, on average, how much does the parlor make per pizza delivery?

Question 12

A certain school is having a graduation ceremony, and graduates are directed onto the stage to receive their diplomas. If the school has graduates and they are called on stage at a Poisson rate of per minute, what is the expected value and standard deviation of the time until all of the graduates are called on stage?

Question 13

Suppose that a certain can is manufactured at a factory in Dayton, OH. The radius of the can is a normal random variable with mean .2cm and Variance .25cm. Find the probability distribution function of the radius of the can.

Question 14

Suppose that the net worth of a member of an audience at a certain self-help seminar is a normal random variable with mean $145,000 and standard deviation $2500. Find the probability that in a group of 14 randomly selected audience members, at least 5 have net worth between $142,500 and $15,0000.