Question

Part 6 of 22 - PDF from complementary CDF - Version 1 Question 6 of 22 5.0 Points Random variable X satisfies PIXSH-forso 1--220 What is fx(-2) O A. exp (4) OB. 1 OC. ATOAAW D. exp(2) O E.O Reset Selection Mark for Review What's This?

Answer 1

Given that the random variable X satisfies

$P[X\leq x]$ is equal to

$1-e^{-2x^{2}}$ when $x\geq 0$ .

and otherwise.

So, this is nothing but the Cumulative probability distribution of the random variable X.

Now, to get the probability density function from the given CDF, we have to differentiate.

So, by applying chain rule, the probability density function will be equal to

$x(x) = (1-2-22

ie. $f_{X}(x)=-\frac{\mathrm{d} }{\mathrm{d} x}(e^{-2x^{2}})$

ie. $f_{X}(x)=-e^{-2x^{2}}\frac{\mathrm{d} }{\mathrm{d} x}(-2x^{2})$

ie. $f_{X}(x)=-e^{-2x^{2}}(-4x)$

ie. $f_{X}(x)=e^{-2x^{2}}(4x)$

So, $f_{X}(x)=e^{-2x^{2}}(4x)$ when $x\geq 0$

and $f_{X}(x)=0$ when

Now, to find $f_{X}(-2).$

As -2 is less than 0, so $f_{X}(-2)$ must be 0.

This can also be intuitively said from the CDF; as there is no area of the curve below the value 0, the probability at -2 must be 0.

So, the correct answer is (E) 0.