Adding two random variables together

Foundations of Probability in R

David Robinson

Chief Data Scientist, DataCamp

Adding two random variables

$$X \sim \text{Binom}(10, .5)$$

$$Y \sim \text{Binom}(100, .2)$$

$$Z \sim X + Y$$

X <- rbinom(100000, 10, .5)
mean(X)
# [1] 5.00938

Y <- rbinom(100000, 100, .2)
mean(Y)
# [1] 19.99422

Z <- X + Y
mean(Z)
# [1] 25.0036

$$E[X+Y]=E[X]+E[Y]$$

X <- rbinom(100000, 10, .5)
var(X)
# [1] 2.500895

Y <- rbinom(100000, 100, .2)
var(Y)
# [1] 16.06289

Z <- X + Y
var(Z)
# [1] 18.58055

$$\text{Var}[X+Y]=\text{Var}[X]+\text{Var}[Y]$$

$$E[X+Y]=E[X]+E[Y]$$

$$\text{ (Even if X and Y aren't independent)}$$

$$\text{Var}[X+Y]=\text{Var}[X]+\text{Var}[Y]$$

$$\text{ (Only if X and Y are independent)}$$

Foundations of Probability in R