Transforming Random Variables

Suppose random variable X represents the number of times a coin lands on heads after three flips. 

In a previous section we derived the probability distribution of X:

The mean of X is μ = 1.5 and the standard deviation is 

Suppose you play a game where you get to flip three coins at once and you get paid $10 for each coin that lands on heads, but the cost to play the game is $5. Let random variable Y represent your expected profit from playing this game. What is the mean (μ) and standard deviation (

To find the mean and standard deviation of Y, we need to apply a linear transformation to random variable X. A linear transformation is simply some change we apply to an existing random variable.

The mean of random variable Y is μy = mμx + b

where μx is the mean of X, m is some number we multiply X by, and b is some number we add to X.

The standard deviation of random variable Y is 

where is some number we multiply by and

In this example, here is how to find the mean of Y:

μy = mμx + b

μy = 10(1.5) + (-5) = 10

We multiply the mean of X by 10 because we earn $10 each time the coin lands on heads, and we subtract 5 because we have to pay $5 just to play the game. Thus, our expected profit from playing this game is $10.

And here’s how to find the standard deviation of Y:

Suppose you play the same game where you get to flip three coins at once. This time, you only get paid $2 for each coin that lands on heads, but you get paid $5 just to play the game. Let random variable Y represent your expected profit from playing this game. What is the mean (μ) and standard deviation (

Here is how to find the mean of Y:

μy = mμx + b

μy = 2(1.5) + (5) = 8

We multiply the mean of X by 2 because we earn $2 each time the coin lands on heads, and we add 5 because we get paid $5 just to play the game. Thus, our expected profit from playing this game is $8.

And here’s how to find the standard deviation of Y:

Suppose you play the same game where you get to flip three coins at once. This time, you get paid $5 for each coin that lands on heads and the game is completely free to play. Let random variable Y represent your expected profit from playing this game. What is the mean (μ) and standard deviation (

Here is how to find the mean of Y:

μy = mμx + b

μy = 5(1.5) = 7.5

We multiply the mean of X by 5 because we earn $5 each time the coin lands on heads, and we don’t add anything for b because the game is free to play. Thus, our expected profit from playing this game is $7.50.

And here’s how to find the standard deviation of Y:

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