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Random Variable

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Random Variable

Study Guide

Key Definition

A random variable assigns a numerical value to each outcome in a sample space. For discrete random variables, the sum of the probabilities of all values equals 1. For continuous random variables, the integral of the probability density function over the entire range equals 1.

Important Notes

The sum of probabilities for all outcomes of a discrete random variable equals 1.
Discrete random variables take countable values.
Continuous random variables take an infinite number of values within a range.
Notation $P(X = x)$ represents the probability that random variable $X$ equals $x$.
Example: Tossing a coin twice can have 0, 1, or 2 tails.

Mathematical Notation

$P(X = x)$ is the probability that random variable $X$ equals $x$

$\frac{1}{4}$ represents one-fourth probability

Remember to use proper notation when solving problems

Why It Works

Random variables allow us to quantify uncertain outcomes using <a href="probability-distribution">$probability distributions$</a>.

Remember

Probability distributions must sum (discrete) or integrate (continuous) to 1.

Quick Reference

Probability of Tails:$P(X = x)$ where $x$ is the number of tails

Understanding Random Variable

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A random variable assigns numbers to outcomes. For example, when flipping a coin twice, X could be the number of heads. We calculate probabilities using simple formulas like P(X = x) = (number of outcomes with value x) ÷ (total number of outcomes).

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

In a coin toss experiment, what is the probability of getting 0 tails if $X$ is the random variable for the number of tails?

Real-World Problem

Question Exercise

Intermediate

Dartboard Scenario

Imagine throwing a dart at a board. If the dartboard has a radius of 9 inches, define a random variable $X$ as the distance in inches from the dart to the center. Then state the range of X and whether X is discrete or continuous.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider a dice roll. Define the random variable $Y$ as the square of the number rolled.

Challenge Quiz

Single Choice Quiz

Advanced

Which of the following correctly describes a continuous random variable?

Recap

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Review key concepts and takeaways