#Transforming and Combining Random Variables 📊

Hey there, future AP Stats pro! Let's dive into transforming and combining random variables. This is a super useful skill that'll pop up all over the exam. Think of it as your secret weapon for simplifying calculations and making sense of data. Let's get started!

#Linear Transformations of a Random Variable

#Adding or Subtracting a Constant ➕➖

When you add or subtract a constant from a random variable, you're basically shifting the entire distribution along the number line. This affects the center and location but not the spread or the shape.

Mean: The mean shifts by the same constant. If Y = X + c, then E(Y) = E(X) + c.
Standard Deviation: The standard deviation stays the same. SD(Y) = SD(X).

#Multiplying or Dividing by a Constant ✖️➗

Multiplying or dividing by a constant changes the center, location, and spread of the distribution but not the shape.

Mean: The mean is multiplied or divided by the same constant. If Y = c * X, then E(Y) = c * E(X).
Standard Deviation: The standard deviation is also multiplied or divided by the same constant. SD(Y) = |c| * SD(X).

Key Concept

Key Point: Remember, adding or subtracting only shifts the center, while multiplying or dividing affects both center and spread. The shape of the distribution remains the same.

Y = a + BX

#Combining Random Variables

Sometimes, you'll need to combine two or more random variables. Here's how to handle it:

#Expected Value of the Sum/Difference of Two Random Variables

#💡 Summary

Sum: If S = X + Y, then E(S) = E(X) + E(Y). The mean of the sum is the sum of the means.
Difference: If D = X - Y, then E(D) = E(X) - E(Y)....

Combining Random Variables

Isabella Lopez