Mean and standard deviation problem

[snoggerT]

The mean and standard deviation of a random variable x are -11 and 4 respectively. Find the mean and standard deviation of the given random variables:

1) y=x+7

2) v=8x

3) w=8x+7



2. Homework Equations : E(x) = u, E(ax+b) = aE(x)+b

The Attempt at a Solution

I've gotten the standard deviations for these problems, though I'm still not sure why they are what they are. I can't figure out how to find the mean though. Can somebody please explain all of this to me? thanks.

 

[LCKurtz]

If you know E(X) = -11, what do your relevant equaitons tell you about E(X + 7)?

 

[snoggerT]

If you know E(X) = -11, what do your relevant equaitons tell you about E(X + 7)?

- I get it. Can you explain the standard deviation portion for me? for instance, when v=8x, the standard deviation is 8*4, but I'm not sure why. Is x the standard deviation in the equation?

 

[Mark44]

When you're dealing with the mean of a transformed random variable X, you can use expectation, E(...), to find the mean of the transformed variable. When you're dealing with the standard deviation, you need something else, Var(...), or variance of a random variable. As you probably know, the variance is the square of the standard deviation, or equivalently, the standard deviation is the square root of the variance.

Since you are asked about the mean and standard deviation of a transformed r.v., I'm going to assume you have been exposed to this concept.

In my book on mathematical statistics, there is a theorem that says:

Let X be a random variable and let a and b be constants. Define Y = aX + b. Then
Var(Y) = a²Var(X)​

Elsewhere in my text Var(X) is defined as E( (X - mu)² ), which turns out to be equal to E(X²) - mu².

In the problem, V = 8X, what would be Var(V)? Further, what would be the standard deviation of V?