GATE CSE 2011 | Question: 18

Question

If the difference between the expectation of the square of a random variable $\left(E\left[X^2\right]\right)$ and the square of the expectation of the random variable $\left(E\left[X\right]\right)^2$ is denoted by $R$, then

• A. $R=0$
• B. $R<0$
• C. $R\geq 0$
• D. $R > 0$

Comments

Catch is that Variance can never be negative.

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Unless we are dealing with numbers with imaginary parts!

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The variance of a random variable X is defined to be Var(X) = E [(X − E [X])]^2 = E[X^2] − E [X] ^2

The variance is always nonnegative since we take the expectation of a nonnegative random variable (X − E [X])^2

source https://web.stanford.edu/class/archive/cs/cs109/cs109.1218/files/student_drive/3.3.pdf see definition 3.3.1

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Variance Concept : [Source : Wikipedia]

https://en.wikipedia.org/wiki/Variance

Answer 1

Answer is (C).
The difference between  $(E[X^2])$ and $(E[X])^{2}$ is called variance of a random variable. Variance measures how far a set of numbers is spread out. (A variance of zero indicates that all the values are identical.) A non-zero variance is always positive.

Comments

I totally agree with the answer. However the term used here is difference. does it not mean it could be mean^2-m.s.v as well.

Answer 2

answer is option c .

Answer 3

V(x) = E(x^2) - [E(x)]² = R

where V(x) is the Variance of x, Since Variance is Square and Hence Never be Neagtive, R>=0

Answer 4

This the definition definition of variance .  Variance never be negative.Variance is the average squared deviation from the mean. Notice the word “squared”. It may be zero.  Variance of constant is zero.Theorem 2 a

Answer C

https://www.macroption.com/can-variance-be-negative/

Answer 5

To get an idea about variance of a random variable.

Answer 6

The difference between  (E[X²]) and (E[X])² is called variance of a random variable.  Variance measures how far a set of numbers is spread out.