Question

If the variance of the random variable X is 5, then the variance of the random variable -3X is
A. 15
B. 45
C. -45
D. 60

Answer 1

Hint:In order to find the solution of this question, we should have a basic knowledge about the properties of variance and therefore we will be able to find the solution of this question. We will use the formula, $ \operatorname{var}\left( cX \right)={{c}^{2}}\operatorname{var}\left( X \right) $ and substitute c = -3, to get the answer.

Complete step-by-step answer:
In this question, we have been given that the variance of the random variable X is 5 and we have been asked to find the variance of the random variable -3X.
Now, we know that the variance follows a few properties which will help us to find the answer of this question. We know that $ \operatorname{var}\left( cX \right)={{c}^{2}}\operatorname{var}\left( X \right) $ , where c represents a constant. So, we can say that if var(X) is given as 5, then variance of -3X will be given by using the above property.
So, let us consider c = -3. So, we can say that,
 $ \operatorname{var}\left( -3X \right)={{\left( -3 \right)}^{2}}\operatorname{var}\left( X \right) $
Now, we know that $ {{\left( -3 \right)}^{2}}=9 $ . So, we can say that,
 $ \operatorname{var}\left( -3X \right)=9\operatorname{var}\left( X \right) $
Now, we will put the value of var(X) as 5. So, we will get,
 $ \operatorname{var}\left( -3X \right)=9\times 5 $
And we know that it can be further written as,
 $ \operatorname{var}\left( -3X \right)=45 $
Here, we can see that var(-3X) is 45 which matches with option (B). Hence, option (B) is the correct answer.

Note: While solving this question, the possible mistake which we can make is by choosing option (c) as the correct answer, but it is wrong. Also, we need to remember that if var(X) = y, then $ \operatorname{var}\left( cX \right)={{c}^{2}}y $ , because this is the base property to solve this question. If we don’t remember this and use var(c) = cy, then the whole solution will become wrong.