PYQs Based Test - 19 : Uniform and Exponential Distribution

Consider that X and Y are independent continuous valued random variables with uniform PDF given by X ~ U(2, 3) and Y ~ U(1, 4). Then P(Y ≤ X) is equal to __________ (rounded off to two decimal places).

If f(𝑥) and g(𝑥) are two probability density functions,

f≤ft( x ) = ≤ft\{ {*20c x/a + 1&: - a ≤ x < 0\\ - x/a + 1&0 ≤ x ≤ a\\ 0&otherwise } .\\ g≤ft( x ) = ≤ft\{ {*20c - x/a&:-a ≤ x ≤ 0\\ x/a&:0 ≤ x ≤ a\\ 0&:otherewise } .

Which one of the following statements is true?

The probability density function of a random variable X is p${}_X$ (x) = e${}^{-x}$ for x ≥ 0 and 0 otherwise. The expected value of the function g${}_X$ (x) = e${}^{3x/4}$ is ___________.

The probability density function of a random variable, xis

$f\left(x\right)=\frac{x}{4}\left(4-x^2\right)$ for $0\le x\le 2$

= 0 otherwise

The mean, μₓ of the random variable is ____.

Let the probability density function of a random variable, $X$, be given as:

$fx\left(x\right)=\frac{3}{2}{e}^{-3x}u\left(x\right)+a{e}^{4x}u\left(-x\right)$

Where $u\left(x\right)$ is the unit step function.

Then the value of 'a' and $Prob\left\{X\le 0\right\}$ respectively, are