Question Type 3: Finding the probability for a given PDF Bootcamps

For the same pdf $f(x)=4e^{-4x}$, $x>0$, find $P(X\le3)$.

Let $X$ be a continuous random variable with pdf $f(x)=4e^{-4x}, \text{ for }x>0.$ Calculate $P(X>12)$.

Determine the expected value $E(X)$ for $X$ with pdf $f(x)=4e^{-4x}$, $x>0$.

With pdf $f(x)=4e^{-4x}$ for $x>0$, compute $P(2<X<5)$.

Compute the variance $\mathrm{Var}(X)$ when $X\sim\mathrm{Exp}(4)$.

Find the median $m$ of $X$ with pdf $f(x)=4e^{-4x}$, $x>0$, such that $P(X\le m)=0.5$.

Find the 90th percentile $x_{0.9}$ of $X$ with pdf $f(x)=4e^{-4x}$, $x>0$.

Verify the memoryless property by showing $P(X>7\mid X>3)=P(X>4)$ for $X\sim\mathrm{Exp}(4)$.

For a random variable with pdf $f(x)=\lambda e^{-\lambda x}$, $x>0$, find $\lambda$ such that $P(X<2)=0.8$.

Let $Y=3X$ where $X\sim\mathrm{Exp}(4)$. (a) Find the pdf of $Y$. (b) Then compute $P(Y>10)$.

Find the conditional pdf of $X$ given $X>4$ if $X\sim\mathrm{Exp}(4)$.

Given $f(x)=5e^{-5x}$ for $x>0$, the hazard function is $h(x)=f(x)/(1-F(x))$. Compute $h(1)$.