Calculating the Variance of 5 - 2X When SD of X is 3

In this article, we explore how to calculate the variance of the transformed random variable Y 5 - 2X given that the standard deviation (SD) of X is 3. We will use properties of variance and explain the underlying reasoning to ensure a deep understanding of the topic.

Variance of a Linear Transformation

To find the variance of a linearly transformed random variable, we use the following property:

For a random variable Y aX b, the variance of Y is given by: Var(Y) a2 middot; Var(X) Where a and b are constants, and Var(X) is the variance of X.

Given Information

We are given:

The standard deviation (SD) of X is 3.

The variance Var(X) can be calculated as:

Var(X) (SD(X))^2 3^2 9

Applying the Linear Transformation

In our case, we have:

a -2 b 5

So, the transformed random variable Y 5 - 2X has variance:

Var(Y) (-2)2 middot; Var(X) 4 middot; 9 36

Therefore, the variance of Y 5 - 2X is 36.

Verification Using Expected Value

To further validate the result, let's use the definition of variance in terms of the expected value operator E.

We start with the definition of variance for a random variable X with mean μ: Var(X) E[(X - μ)2] (SD(X))^2 9 Let U 5 - 2X. We can write: U - μU -2(X - μ)

Therefore:

E[(U - μU)2] E[(-2)(X - μ)2] 4 middot; E[(X - μ)2]

Since E[(X - μ)2] Var(X), we get:

Var(U) 4 middot; Var(X) 4 middot; 9 36

Alternate Approach: Properties of Variance and Standard Deviation

An alternative, straightforward method involves using the properties of variance and standard deviation:

We know that the standard deviation of X is 3. The variance of X is (SD(X))^2 3^2 9. Addition or subtraction of a constant (5 in this case) does not affect the variance. The formula for the variance of a scaled random variable is: Var(ax b) a2 middot; Var(x)

Given that a -2, we use the formula:

Var(5 - 2X) (-2)2 middot; Var(X) 4 middot; 9 36

Thus, the variance of Y 5 - 2X is 36.

Conclusion

In conclusion, by using the properties of variance and the given standard deviation, we have calculated the variance of the transformed random variable 5 - 2X. The variance is 36, and this result can be verified through multiple methods as discussed in this article.