Register

GO Classes GATE DA | Probability | Weekly Quiz 8 | Question: 10

• retagged by
358 views

1 Answer

4 4 votes
Solution. We have

$$
E(|X|)=\int_{-\infty}^{\infty}|x| \frac{1}{4} e^{-|x| / 2} \mathrm{~d} x=2 \int_{0}^{\infty} x \frac{1}{4} e^{-x / 2} \mathrm{~d} x=\int_{0}^{\infty} x \frac{1}{2} e^{-x / 2} \mathrm{~d} x=2
$$

where the last equality follows by observing that the integral is nothing but the expected value of an exponentially distributed random variable with mean 2 .
Voting & Accepting an answer helps improve the community
Answer:
← Previous · GO Classes GATE DA |...Next → · GO Classes GATE DA |...
← Previous in category · GO Classes GATE DA |...Next in category → · GO Classes GATE DA |...
Position:
Show:

Related questions

0 0 votes
1 1 answer
261
261 views
GO Classes asked Jun 27, 2025
261 views
GO Classes GATE DA | Probability | Weekly Quiz 8 | Question: 9
The probability density function of a random variable $X$ is given by$$f(x)= \begin{cases}\frac{1}{4}, & \text { if }|x| \leq 1 \\ \frac{1}{4 x^{2}}, & \text { otherwise ...
3 3 votes
2 2 answers
376
376 views
GO Classes asked Jun 27, 2025
376 views
GO Classes GATE DA | Probability | Weekly Quiz 8 | Question: 11
Let $Y \sim U(0,1)$. The conditional probability density function of $X$ given $Y$ is$$f_{X \mid Y}(x \mid y)= \begin{cases}\frac{1}{y}, & \text { if } 0<x<y<1 \\ 0, & \t...
2 2 votes
1 1 answer
347
347 views
GO Classes asked Jun 27, 2025
347 views
GO Classes GATE DA | Probability | Weekly Quiz 8 | Question: 12
Let $X$ be a continuous random variable defined on $[0,1]$ such that its probability density function $f(x)=1$ for $0 \leq x \leq 1$ and 0 otherwise. Let $Y=\log _{\mathr...
2 2 votes
1 1 answer
306
306 views
GO Classes asked Jun 27, 2025
306 views
GO Classes GATE DA | Probability | Weekly Quiz 8 | Question: 13
Let $X$ and $Y$ be independent standard normal random variables, that is, $X, Y \sim \mathcal{N}(0,1)$. Define$$U=\frac{X}{Y}, \quad V=\frac{X}{|Y|}$$Then the value of $\...