Exam P Distributions From Problems

A fair six-sided die is rolled once. Let X be the number shown.

Discrete Uniform — Discrete — P(X=x)=1/6, x=1,2,…,6

A random integer is selected from 1 through 10 with each integer equally likely. Let X be the selected integer.

Discrete Uniform — Discrete — P(X=x)=1/10, x=1,2,…,10

A spinner has 8 equal sections numbered 1 through 8. Let X be the number landed on.

Discrete Uniform — Discrete — P(X=x)=1/8, x=1,2,…,8

A card numbered 1 through 20 is selected randomly, with all cards equally likely. Let X be the number selected.

Discrete Uniform — Discrete — P(X=x)=1/20, x=1,2,…,20

A computer randomly generates one integer from 0 through 9. Let X be the generated integer.

Discrete Uniform — Discrete — P(X=x)=1/10, x=0,1,…,9

A fair 12-sided die numbered 1 through 12 is rolled. Let X be the result.

Discrete Uniform — Discrete — P(X=x)=1/12, x=1,2,…,12

One of 5 equally likely prizes numbered 1 through 5 is selected. Let X be the prize number.

Discrete Uniform — Discrete — P(X=x)=1/5, x=1,2,…,5

A random integer is selected from -3 through 3, with all values equally likely. Let X be the result.

Discrete Uniform — Discrete — P(X=x)=1/7, x=-3,-2,-1,0,1,2,3

A basketball player makes a free throw with probability 0.80. Let X=1 if the player makes one attempted free throw and X=0 otherwise.

Bernoulli — Discrete — P(X=x)=(0.8)^x(0.2)^(1-x), x=0,1

A product is defective with probability 0.05. Let X=1 if a randomly selected product is defective and X=0 otherwise.

Bernoulli — Discrete — P(X=x)=(0.05)^x(0.95)^(1-x), x=0,1

A student has a 0.70 probability of answering one question correctly. Let X=1 for correct and X=0 for incorrect.

Bernoulli — Discrete — P(X=x)=(0.7)^x(0.3)^(1-x), x=0,1

A coin has probability 0.50 of landing heads. Let X=1 for heads and X=0 for tails.

Bernoulli — Discrete — P(X=x)=(0.5)^x(0.5)^(1-x), x=0,1

A customer has a 0.30 probability of making a purchase. Let X=1 if the customer purchases and X=0 otherwise.

Bernoulli — Discrete — P(X=x)=(0.3)^x(0.7)^(1-x), x=0,1

A machine successfully completes a task with probability 0.95. Let X=1 for success and X=0 for failure.

Bernoulli — Discrete — P(X=x)=(0.95)^x(0.05)^(1-x), x=0,1

A randomly selected voter supports a proposal with probability 0.60. Let X=1 for support and X=0 otherwise.

Bernoulli — Discrete — P(X=x)=(0.6)^x(0.4)^(1-x), x=0,1

An email has a 0.10 probability of being spam. Let X=1 if the email is spam and X=0 otherwise.

Bernoulli — Discrete — P(X=x)=(0.1)^x(0.9)^(1-x), x=0,1

A basketball player makes each free throw independently with probability 0.80. Let X be the number made in 10 attempts.

Binomial — Discrete — P(X=x)=C(10,x)(0.8)^x(0.2)^(10-x)

A fair coin is flipped 20 times. Let X be the number of heads.

Binomial — Discrete — P(X=x)=C(20,x)(0.5)^x(0.5)^(20-x)

Each product has a 0.05 probability of being defective. Let X be the number of defective products among 30 independently selected products.

Binomial — Discrete — P(X=x)=C(30,x)(0.05)^x(0.95)^(30-x)

A student has a 0.70 probability of answering each question correctly. Let X be the number correct out of 15 independent questions.

Binomial — Discrete — P(X=x)=C(15,x)(0.7)^x(0.3)^(15-x)

Each customer independently has a 0.25 probability of purchasing a product. Let X be the number who purchase among 12 customers.

Binomial — Discrete — P(X=x)=C(12,x)(0.25)^x(0.75)^(12-x)

A baseball player gets a hit with probability 0.30 on each independent at-bat. Let X be the number of hits in 8 at-bats.

Binomial — Discrete — P(X=x)=C(8,x)(0.3)^x(0.7)^(8-x)

A website visitor clicks an advertisement with probability 0.10 independently. Let X be the number of clicks among 50 visitors.

Binomial — Discrete — P(X=x)=C(50,x)(0.1)^x(0.9)^(50-x)

A medication works for each patient independently with probability 0.75. Let X be the number for whom it works among 20 patients.

Binomial — Discrete — P(X=x)=C(20,x)(0.75)^x(0.25)^(20-x)

A six-sided die is rolled 24 times. Let X be the number of times a 6 appears.

Binomial — Discrete — P(X=x)=C(24,x)(1/6)^x(5/6)^(24-x)

A basketball player makes each free throw independently with probability 0.80. Let X be the number of attempts needed to make the first shot.

Geometric — Discrete — P(X=x)=(0.2)^(x-1)(0.8), x=1,2,…

A fair coin is repeatedly flipped. Let X be the number of flips needed to get the first head.

Geometric — Discrete — P(X=x)=(0.5)^(x-1)(0.5), x=1,2,…

A salesperson makes a sale on each independent call with probability 0.20. Let X be the number of calls needed to make the first sale.

Geometric — Discrete — P(X=x)=(0.8)^(x-1)(0.2), x=1,2,…

A machine produces a defective item with probability 0.05 independently. Let X be the number of items inspected until the first defective item.

Geometric — Discrete — P(X=x)=(0.95)^(x-1)(0.05), x=1,2,…

A student guesses on questions and has probability 0.25 of answering each correctly. Let X be the number of questions attempted until the first correct answer.

Geometric — Discrete — P(X=x)=(0.75)^(x-1)(0.25), x=1,2,…

A six-sided die is repeatedly rolled. Let X be the number of rolls required to get the first 6.

Geometric — Discrete — P(X=x)=(5/6)^(x-1)(1/6), x=1,2,…

A customer purchases with probability 0.30 on each independent visit. Let X be the number of visits until the first purchase.

Geometric — Discrete — P(X=x)=(0.7)^(x-1)(0.3), x=1,2,…

A component passes inspection with probability 0.90 independently. Let X be the number inspected until the first component passes.

Geometric — Discrete — P(X=x)=(0.1)^(x-1)(0.9), x=1,2,…

A box contains 20 red balls and 80 blue balls. Ten balls are selected without replacement. Let X be the number of red balls selected.

Hypergeometric — Discrete — P(X=x)=[C(20,x)C(80,10-x)]/C(100,10)

A class has 15 seniors and 35 juniors. Eight students are randomly selected without replacement. Let X be the number of seniors selected.

Hypergeometric — Discrete — P(X=x)=[C(15,x)C(35,8-x)]/C(50,8)

A shipment contains 10 defective and 40 nondefective products. Five are selected without replacement. Let X be the number of defective products selected.

Hypergeometric — Discrete — P(X=x)=[C(10,x)C(40,5-x)]/C(50,5)

A deck contains 4 aces and 48 non-aces. Five cards are drawn without replacement. Let X be the number of aces drawn.

Hypergeometric — Discrete — P(X=x)=[C(4,x)C(48,5-x)]/C(52,5)

A population contains 30 people who support a proposal and 70 who do not. Twelve are sampled without replacement. Let X be the number who support it.

Hypergeometric — Discrete — P(X=x)=[C(30,x)C(70,12-x)]/C(100,12)

A warehouse contains 25 premium products and 75 standard products. Twenty products are sampled without replacement. Let X be the number of premium products selected.

Hypergeometric — Discrete — P(X=x)=[C(25,x)C(75,20-x)]/C(100,20)

A group contains 8 trained employees and 22 untrained employees. Six are randomly chosen without replacement. Let X be the number of trained employees selected.

Hypergeometric — Discrete — P(X=x)=[C(8,x)C(22,6-x)]/C(30,6)

A batch contains 12 damaged items and 48 undamaged items. Ten are selected without replacement. Let X be the number of damaged items selected.

Hypergeometric — Discrete — P(X=x)=[C(12,x)C(48,10-x)]/C(60,10)

A basketball player makes each free throw with probability 0.70. Let X be the total number of attempts required to make the 3rd free throw.

Negative Binomial — Discrete — P(X=x)=C(x-1,2)(0.7)^3(0.3)^(x-3), x=3,4,…

A salesperson makes a sale with probability 0.20 on each independent call. Let X be the total number of calls required to make the 5th sale.

Negative Binomial — Discrete — P(X=x)=C(x-1,4)(0.2)^5(0.8)^(x-5), x=5,6,…

A fair coin is repeatedly flipped. Let X be the total number of flips required to obtain the 4th head.

Negative Binomial — Discrete — P(X=x)=C(x-1,3)(0.5)^4(0.5)^(x-4), x=4,5,…

A student answers each question correctly with probability 0.60. Let X be the number of questions attempted to obtain the 6th correct answer.

Negative Binomial — Discrete — P(X=x)=C(x-1,5)(0.6)^6(0.4)^(x-6), x=6,7,…

A machine produces a defective item with probability 0.10. Let X be the total number produced until the 2nd defective item occurs.

Negative Binomial — Discrete — P(X=x)=C(x-1,1)(0.1)^2(0.9)^(x-2), x=2,3,…

A baseball player gets a hit with probability 0.30 per at-bat. Let X be the total number of at-bats needed to get the 5th hit.

Negative Binomial — Discrete — P(X=x)=C(x-1,4)(0.3)^5(0.7)^(x-5), x=5,6,…

A six-sided die is repeatedly rolled. Let X be the total number of rolls required to obtain the 3rd six.

Negative Binomial — Discrete — P(X=x)=C(x-1,2)(1/6)^3(5/6)^(x-3), x=3,4,…

A component passes inspection with probability 0.80. Let X be the number inspected to obtain the 4th passing component.

Negative Binomial — Discrete — P(X=x)=C(x-1,3)(0.8)^4(0.2)^(x-4), x=4,5,…

A call center receives an average of 5 calls per hour. Let X be the number of calls received in one hour.

Poisson — Discrete — P(X=x)=e^(-5)5^x/x!, x=0,1,2,…

A website receives an average of 10 visits per minute. Let X be the number of visits in one minute.

Poisson — Discrete — P(X=x)=e^(-10)10^x/x!, x=0,1,2,…

A road has an average of 2 accidents per month. Let X be the number of accidents in one month.

Poisson — Discrete — P(X=x)=e^(-2)2^x/x!, x=0,1,2,…

A machine experiences an average of 0.5 failures per day. Let X be the number of failures in one day.

Poisson — Discrete — P(X=x)=e^(-0.5)(0.5)^x/x!, x=0,1,2,…

A store receives an average of 8 customers every 10 minutes. Let X be the number arriving during a 10-minute interval.

Poisson — Discrete — P(X=x)=e^(-8)8^x/x!, x=0,1,2,…

A book contains an average of 3 printing errors per page. Let X be the number of errors on one page.

Poisson — Discrete — P(X=x)=e^(-3)3^x/x!, x=0,1,2,…

A hospital averages 12 emergency arrivals per hour. Let X be the number of arrivals in one hour.

Poisson — Discrete — P(X=x)=e^(-12)12^x/x!, x=0,1,2,…

A parking lot receives an average of 4 cars every five minutes. Let X be the number arriving in five minutes.

Poisson — Discrete — P(X=x)=e^(-4)4^x/x!, x=0,1,2,…

A server experiences an average of 1.5 errors per hour. Let X be the number of errors during one hour.

Poisson — Discrete — P(X=x)=e^(-1.5)(1.5)^x/x!, x=0,1,2,…

A bus arrives at a random time between 0 and 10 minutes from now, with every time equally likely. Let X be the waiting time.

Continuous Uniform — Continuous — f(x)=1/10, 0<=x<=10

A machine completes a process at a random time between 2 and 7 minutes, with all times equally likely. Let X be the completion time.

Continuous Uniform — Continuous — f(x)=1/5, 2<=x<=7

A customer arrives at a random time between 0 and 20 minutes, with all times equally likely. Let X be the arrival time.

Continuous Uniform — Continuous — f(x)=1/20, 0<=x<=20

A random number is selected uniformly from 5 to 15. Let X be the selected value.

Continuous Uniform — Continuous — f(x)=1/10, 5<=x<=15

A train arrives uniformly between 3 and 11 minutes from now. Let X be the waiting time.

Continuous Uniform — Continuous — f(x)=1/8, 3<=x<=11

A temperature measurement is equally likely to be any value between 20 and 25 degrees. Let X be the temperature.

Continuous Uniform — Continuous — f(x)=1/5, 20<=x<=25

A delivery occurs at a uniformly random time between 1 and 4 hours from now. Let X be the delivery time.

Continuous Uniform — Continuous — f(x)=1/3, 1<=x<=4

A computer generates a real number uniformly between -5 and 5. Let X be the generated number.

Continuous Uniform — Continuous — f(x)=1/10, -5<=x<=5

The average waiting time until the next customer arrives is 0.5 hour, and waiting times are memoryless. Let X be the waiting time.

Exponential — Continuous — f(x)=2e^(-2x), x>=0

The average lifetime of a component is 4 years, and its lifetime is exponentially distributed. Let X be its lifetime.

Exponential — Continuous — f(x)=0.25e^(-0.25x), x>=0

Customers arrive according to a Poisson process at a rate of 5 per hour. Let X be the waiting time until the next customer.

Exponential — Continuous — f(x)=5e^(-5x), x>=0

Calls arrive according to a Poisson process at a rate of 3 per minute. Let X be the time until the next call.

Exponential — Continuous — f(x)=3e^(-3x), x>=0

A device has an exponentially distributed lifetime with an average lifetime of 2 hours. Let X be its lifetime.

Exponential — Continuous — f(x)=0.5e^(-0.5x), x>=0

A server receives requests according to a Poisson process at a rate of 4 per second. Let X be the waiting time until the next request.

Exponential — Continuous — f(x)=4e^(-4x), x>=0

An event occurs at an average rate of 10 per hour according to a Poisson process. Let X be the waiting time until the next event.

Exponential — Continuous — f(x)=10e^(-10x), x>=0

A machine fails at a constant rate of 0.2 per hour. Let X be the time until failure.

Exponential — Continuous — f(x)=0.2e^(-0.2x), x>=0

Test scores are normally distributed with mean 100 and standard deviation 15. Let X be a randomly selected score.

Normal — Continuous — f(x)=1/(15sqrt(2pi)) * e^(-(x-100)^2/(2(15)^2))

Adult heights are normally distributed with mean 68 inches and standard deviation 3 inches. Let X be a randomly selected height.

Normal — Continuous — f(x)=1/(3sqrt(2pi)) * e^(-(x-68)^2/(2(3)^2))

Package weights are normally distributed with mean 50 pounds and standard deviation 5 pounds. Let X be the weight of a randomly selected package.

Normal — Continuous — f(x)=1/(5sqrt(2pi)) * e^(-(x-50)^2/(2(5)^2))

Battery lives are normally distributed with mean 10 hours and standard deviation 2 hours. Let X be the life of a randomly selected battery.

Normal — Continuous — f(x)=1/(2sqrt(2pi)) * e^(-(x-10)^2/(2(2)^2))

Daily temperatures are normally distributed with mean 75 degrees and standard deviation 8 degrees. Let X be a randomly selected daily temperature.

Normal — Continuous — f(x)=1/(8sqrt(2pi)) * e^(-(x-75)^2/(2(8)^2))

Exam completion times are normally distributed with mean 60 minutes and standard deviation 10 minutes. Let X be a student's completion time.

Normal — Continuous — f(x)=1/(10sqrt(2pi)) * e^(-(x-60)^2/(2(10)^2))

Product lengths are normally distributed with mean 20 cm and standard deviation 0.5 cm. Let X be a product's length.

Normal — Continuous — f(x)=1/(0.5sqrt(2pi)) * e^(-(x-20)^2/(2(0.5)^2))

IQ scores are normally distributed with mean 100 and standard deviation 15. Let X be a randomly selected IQ score.

Normal — Continuous — f(x)=1/(15sqrt(2pi)) * e^(-(x-100)^2/(2(15)^2))

Commute times are normally distributed with mean 30 minutes and standard deviation 6 minutes. Let X be a randomly selected commute time.

Normal — Continuous — f(x)=1/(6sqrt(2pi)) * e^(-(x-30)^2/(2(6)^2))

A random proportion X between 0 and 1 follows a beta distribution with alpha=2 and beta=2.

Beta — Continuous — f(x)=6x(1-x), 0<x<1

A random probability X follows a beta distribution with alpha=2 and beta=4.

Beta — Continuous — f(x)=20x(1-x)^3, 0<x<1

A random proportion X follows a beta distribution with alpha=3 and beta=2.

Beta — Continuous — f(x)=12x^2(1-x), 0<x<1

A random proportion X follows a beta distribution with alpha=1 and beta=3.

Beta — Continuous — f(x)=3(1-x)^2, 0<x<1

A random probability X follows a beta distribution with alpha=5 and beta=1.

Beta — Continuous — f(x)=5x^4, 0<x<1

A random proportion X follows a beta distribution with alpha=3 and beta=3.

Beta — Continuous — f(x)=30x^2(1-x)^2, 0<x<1

A random probability X follows a beta distribution with alpha=2 and beta=6.

Beta — Continuous — f(x)=42x(1-x)^5, 0<x<1

A random proportion X follows a beta distribution with alpha=4 and beta=2.

Beta — Continuous — f(x)=20x^3(1-x), 0<x<1

The waiting time X until the 2nd event in a Poisson process with rate 1 event per minute is recorded.

Gamma — Continuous — f(x)=xe^(-x), x>0

The waiting time X until the 3rd event in a Poisson process with rate 1 event per minute is recorded.

Gamma — Continuous — f(x)=x^2e^(-x)/2, x>0

The waiting time X until the 2nd event in a Poisson process with rate 2 events per minute is recorded.

Gamma — Continuous — f(x)=4xe^(-2x), x>0

The waiting time X until the 3rd event in a Poisson process with rate 3 events per hour is recorded.

Gamma — Continuous — f(x)=27x^2e^(-3x)/2, x>0

The waiting time X until the 4th event in a Poisson process with rate 2 events per hour is recorded.

Gamma — Continuous — f(x)=16x^3e^(-2x)/6, x>0

The waiting time X until the 2nd event in a Poisson process with rate 5 events per hour is recorded.

Gamma — Continuous — f(x)=25xe^(-5x), x>0

The waiting time X until the 5th event in a Poisson process with rate 1 event per hour is recorded.

Gamma — Continuous — f(x)=x^4e^(-x)/24, x>0

The waiting time X until the 3rd event in a Poisson process with rate 2 events per hour is recorded.

Gamma — Continuous — f(x)=4x^2e^(-2x), x>0