Unit 3: Random Variables and Their Distributions
Turning outcomes into numbers, and the discrete distribution family — Bernoulli through Poisson — that a huge share of real-world randomness reduces to.
11 topics · 40 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
3.1 What a Random Variable Actually Is
- How it worksHow does a random variable assign one number to every outcome in the sample space?
- Common mistakeWhy isn't a random variable random at all once the underlying experiment has actually happened?
- Going furtherWhat happens to a random variable's distribution when you apply a function to it?
Your turn1 of 2
A random variable X takes the value -4 or 7, each with probability 0.5. Let Y = X2. Find E[Y].3.2 PMFs and CDFs
- How it worksHow does a PMF list every value's probability while a CDF accumulates them from the left?
- Worked exampleFind the CDF value at x=3 for a random variable with PMF 0.2, 0.3, 0.5 at values 1, 2, 3.
- Common mistakeWhy isn't a CDF required to jump at every integer just because its PMF is discrete?
- Going furtherWhat happens to a CDF's graph at a value where the PMF assigns zero probability?
Your turn1 of 2
A random variable takes values 1, 2, and 3 with PMF probabilities 0.309, 0.295, and whatever remains. Find the CDF value at x=2 (that is, P(X ≤ 2)).3.2 The Discrete Uniform Distribution
- How it worksHow does a discrete uniform distribution spread probability equally across a finite set of values?
- Worked exampleFind the mean and variance of a discrete uniform distribution on the integers 1 through 10.
- Going furtherWhat happens to a discrete uniform distribution's variance as its range of values doubles?
Your turn1 of 2
Find the mean of a discrete uniform distribution on the integers 1 through 36.3.3 The Bernoulli Distribution
- How it worksHow does one parameter p fully determine a Bernoulli random variable's entire distribution?
- Worked exampleFind the mean and variance of a Bernoulli random variable with p equal to 0.3.
- Common mistakeWhy isn't a Bernoulli random variable the same thing as a coin flip labeled heads or tails?
Your turn1 of 2
Find the variance of a Bernoulli random variable with p = 0.677.3.3 The Binomial Distribution
- How it worksHow does summing 20 independent Bernoulli trials build the binomial distribution's whole shape?
- Worked exampleFind the probability of exactly 3 heads in 10 flips of a fair coin.
- Worked exampleFind the probability a factory with a 2 percent defect rate ships exactly 1 defective item in a box of 20.
- Common mistakeWhy doesn't doubling the number of trials in a binomial distribution double its standard deviation?
- Going furtherWhat happens to a binomial distribution's shape as n grows large while p stays fixed?
Your turn1 of 2
A fair coin is flipped 15 times. Find the probability of exactly 3 heads.3.4 The Hypergeometric Distribution
- How it worksHow does sampling without replacement from a finite population produce the hypergeometric formula?
- Worked exampleFind the probability of drawing exactly 2 red balls in 5 draws without replacement from an urn of 10 red and 15 blue balls.
- Common mistakeWhy isn't the hypergeometric distribution just the binomial distribution with a fixed success probability?
- Ties togetherHow does the hypergeometric distribution turn into the binomial distribution as the population size grows huge?
Your turn1 of 2
An urn has 15 red balls and 19 blue balls. Find the probability of drawing exactly 2 red balls in 5 draws without replacement.3.5 The Geometric Distribution
- How it worksHow does waiting through a string of failures before one success build the geometric PMF?
- Common mistakeWhy doesn't a fair coin that's landed tails 5 times in a row become more likely to land heads next?
- Worked exampleFind the probability that the first six on a fair die appears on exactly the 4th roll.
- Going furtherWhat happens to the geometric distribution's mean as the success probability p shrinks toward zero?
Your turn1 of 2
An event with probability 0.363 occurs independently on each trial. Find the probability that the first occurrence happens on exactly trial number 10.3.6 The Negative Binomial Distribution
- How it worksHow does summing r independent geometric waiting times build the negative binomial distribution?
- Worked exampleFind the probability that the 3rd six on a fair die appears on exactly the 7th roll.
- Going furtherWhat happens to the negative binomial distribution when the number of required successes r equals 1?
Your turn1 of 2
An event with probability 0.309 occurs independently on each trial. Find the probability that the 3rd occurrence happens on exactly trial number 15.3.7 Indicator Random Variables
- How it worksHow does an indicator random variable's expected value collapse to just the probability of its event?
- Worked exampleFind the expected number of fixed points in a random shuffle of 5 people using indicator random variables.
- Ties togetherHow does writing the binomial count as a sum of indicators connect back to linearity of expectation?
Your turn1 of 2
In a random shuffle of 38 people, 7 of them are being specifically tracked. Using indicator random variables, find the expected number of THOSE 7 people who end up in their own original position.3.8 The Poisson Distribution
- How it worksHow does a single rate parameter lambda determine every probability in the Poisson distribution?
- Common mistakeWhy doesn't the Poisson distribution need an upper limit on how many events can occur?
- Worked exampleFind the probability of exactly 3 calls arriving at a call center in an hour when the average rate is 5 calls per hour.
- Going furtherWhat happens to the Poisson distribution's shape as lambda grows from 1 to 20?
Your turn1 of 2
Find the probability of exactly 3 calls arriving at a call center in an hour when the average rate is 11 calls per hour.3.9 The Poisson Approximation to the Binomial
- How it worksHow does letting n grow to infinity while np stays fixed turn the binomial formula into the Poisson formula?
- Worked exampleFind the approximate probability of exactly 2 typos in a 1000-word document if each word has a 0.003 chance of a typo.
- Common mistakeWhy doesn't the Poisson approximation need the individual trials to be identical, only rare and roughly independent?
- Ties togetherHow does the Poisson paradigm connect rare independent events to the birthday problem's collision count?
Your turn1 of 2
Find the approximate probability of exactly 2 typos in a 1545-word document if each word has a 0.00684 chance of a typo (Poisson approximation).