๐Ÿ“š

All Subjects

ย >ย 

๐Ÿ“Šย 

AP Stats

ย >ย 

๐ŸŽฒ

Unit 4

4.9 Combining Random Variables

2 min readโ€ขjune 3, 2020

Kanya Shah


This section is about transforming random variables by adding/subtracting or multiplying/dividing by a constant. Make sure you know how to combine random variables to calculate and interpret the mean and standard deviation.

Linear Transformations of a Random Variable

Y = a + BX

Adding/Subtracting by a constant affects measures of center and location but does NOT affect variability or the shape of a distribution. Multiplying or dividing by a constant affects center, location, and variability measures but wonโ€™t change the shape of a distribution.ย 

How to Find the Expected Value of the Sum/Difference of Two Random Variables

Sum: For any two random variables X and Y, if S = X + Y, the mean of S is meanS= meanX + meanY. Put simply, the mean of the sum of two random variables is equal to the sum of their means.ย 

Difference: For any two random variables X and Y, if D = X - Y, the mean of D is meanD= meanX - meanY. The mean of the difference of two random variables is equal to the difference of their means. The order of subtraction is important.ย 

Independent Random Variables: If knowing the value of X does not help us predict the value of X, then X and Y are independent random variables.ย 

Standard Deviation of the Sum/Difference of Two Independent Random Variables

Sum: For any two independent random variables X and Y, if S = X + Y, the variance of S is SD^2= (X+Y)^2 . To find the standard deviation, take the square root of the variance formula: SD = sqrt(SDX^2 + SDY^2).ย 

Standard deviations do not add; use the formula or your calculator.ย 

Difference: For any two independent random variables X and Y, if D = X - Y, the variance of D is D^2= (X-Y)^2=x2+Y2. To find the standard deviation, take the square root of the variance formula: D=sqrt(x2+Y2). Notice that you are NOT subtracting the variances (or the standard deviation in the latter formula).ย 

๐ŸŽฅWatch: AP Stats - Combining Random Variables

Resources:

Was this guide helpful?

Join us on Discord

Thousands of students are studying with us for the AP Statistics exam.

join now

Browse Study Guides By Unit

โœ๏ธ
Blogs

โœ๏ธ
Free Response Questions (FRQs)

๐Ÿง
Multiple Choice Questions (MCQs)

๐Ÿ‘†
Unit 1: Exploring One-Variable Data

โœŒ๏ธ
Unit 2: Exploring Two-Variable Data

๐Ÿ”Ž
Unit 3: Collecting Data

๐Ÿ“Š
Unit 5: Sampling Distributions

โš–๏ธ
Unit 6: Inference for Categorical Data: Proportions

๐Ÿ˜ผ
Unit 7: Inference for Qualitative Data: Means

โœณ๏ธ
Unit 8: Inference for Categorical Data: Chi-Square

๐Ÿ“ˆ
Unit 9: Inference for Quantitative Data: Slopes

Play this on HyperTyper

Practice your typing skills while reading Combining Random Variables

Start Game