Random variables (RVs)
For a given sample space \(\Omega\) of some experiment, a random variable is defined as a function that assigns a number to each outcome.
Example
Random variable \(X\):
Roll a die 🎲 and record the roll.
Example
Random variable \(X\):
Roll a pair of dice 🎲 🎲 and record the maximum roll.
Is it a random variable?
Roll a pair of dice 🎲 🎲
- \(X\): the sum of the two rolls
- \(Y\): the number of doubles we get
- \(Z\): the second roll raised to the square (e.g., \(3^2=9\))
- \(W\): the sequence of two rolls (e.g., 5 followed by 3)
Discrete random variables
The variable takes finite or countably infinite values.
- Toss a coin 4 times, the number of heads
- Roll a die 🎲 and record the number it lands on
- The number of orders that a shop receives in a day
Continuous random variables
The variable takes infinitely many, uncountable values.
- The speed of riding a bicycle
- The water level of a dam
- The time when customers arrive at a shop
Rule of thumb:
- If you can count it, it is discrete.
- If you need to measure it, it is continuous.
Functions of random variables
\(Y\) is a function of a random variable \(X\)
\[Y=g(X)\]
Is \(Y\) a random variable?
