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

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

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

Consider the probability density function $f(x)=4e^{-4x}$ for $x>0$. Find $P(X \le 3)$.

Let $X$ be a continuous random variable with pdf $f(x)=4e^{-4x}$ for $x>0$. Calculate $P(X>0.5)$.

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

Let $X$ be a continuous random variable with probability density function $f(x) = 4e^{-4x}$ for $x > 0$. Find $P(2 < X < 5)$.

Find the conditional probability density function (PDF) of $X$ given $X > 4$, if $X \sim \text{Exp}(4)$.

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

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

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

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