### Random Variables & Distributions

Life is unpredictable, and probability is our best mathematical handle on real-world uncertainty.

In this quiz, we'll use simple probabilistic models to introduce random variables, a key concept in quantifying chance outcomes.

# Random Variables

A random variable assigns a numerical value to each outcome of a chance event. Random variables are not the same as the events they quantify.

For example, when a fair coin is flipped exactly twice, the set of all outcomes, or sample space, $\Omega$ is $\{ HH, HT, TH, TT \},$ where H/T is short for “heads/tails”, HH is the event the coin lands heads up on both flips, HT represents a heads followed by a tails, etc.

HH, HT, TH, TT are not random variables, but the number of heads observed in two flips is. What are some other examples of random variables?

# Random Variables

Select one or more

Let's take a closer look at random variables through an example.

Let $\Omega = \{ H, T \}$ be the sample space for a single fair coin flip. H and T are not random variables, but if we assign the number 1 to H and the number 0 to T, then we've just created one: $X(H) = 1, \ X(T) = 0.$ $X$ counts the number of heads in a single coin flip.

We don't know what $X$ will be after a flip, but we can compute $P(X=1),$ the probability that we observe $X = 1$ upon flipping the coin. What is $P(X=1) ?$

# Random Variables

Let's make things more interesting by flipping the fair coin three times consecutively.

$N,$ the number of tails in three flips, assigns a value to each element in the sample space $\Omega:$ $\Omega = \{\small{ HHH, HHT, HTT, TTT, THH,TTH, HTH, THT} \}.$ For example, $N(HTT) = 2.$ Since fair coin flips are unpredictable, $N$ is a random variable.

Calculate $P(N = 2).$

# Random Variables

Two dice are rolled, and we let $X$ be the sum of the two values that turn up. Find $P( X \geq 10).$ $\\\\$ A dice roll with $X = 10$

# Random Variables

Functions create new random variables out of more basic ones.

For example, say Xavier, Yoshi, and Zander all take a multiple-choice test by guessing; their scores are the random variables X, Y, and Z. The average $A = \frac{1}{3} ( X + Y + Z )$ is a function of the original three random variables and is a brand-new random variable on its own.

If the exam has three true/false questions and a score less than $20 \%$ doesn't pass, what is the probability the group of Xavier, Yoshi, and Zander fails on average?

# Random Variables

So far, all of our random variables have been discrete, meaning their values are countable.

There are many real-world problems best modeled by a continuum of values; we associate to them continuous random variables.

For example, the velocity $V$ of an air molecule inside of a basketball can take on a continuous range of values. We can't know for sure what it is, so $V$ is a continuous random variable. Air molecules moving about with random velocities drawn from a continuum of possibilities

What are some other continuous random variables?

# Random Variables

Select one or more

Let's say a number $x$ is drawn from the interval $[0,1]$ in such a way that if $0 \leq a , b \leq 1,$ then $P( a \leq x \leq b ) = b-a.$ Define the random variable $Y = 2 x + 5.$ What is $P( 6 \leq Y \leq 7) ?$

# Random Variables

Since the number of outcomes is uncountable for a continuous random variable, we need to take a different approach to how we compute its probabilities.

We'll see in the next quiz that dealing with continuous random variables is relatively straightforward if we have a probability distribution.

# Random Variables

×