Question Type 6: Finding the expected value for a given continuous distribution and PDF Exercises

A random variable $X$ has pdf $f(x)=ke^{-x/2}$ for $x>0$. Find $k$ and $E(X)$.

Let $X$ have pdf $f(x)=\frac{3}{8}x^2$ for $0\le x\le2$. Compute the expected value $E(X)$.

If $X$ has probability density function $f(x)=6x(1-x)$ for $0<x<1$, find the expected value $E(X)$.

For a random variable $X$ with probability density function (pdf) $f(x)=\frac{3}{4}(1-x^2)$ for $-1 \leq x \leq 1$, find $E(X)$.

The pdf of $X$ is given by $f(x)=kx$ for $0\le x\le2$, and 0 otherwise. Determine $k$ and find $E(X)$.

Consider the pdf $f(x)=\frac{c}{x^4}$ for $x\ge1$. Find $c$ and evaluate whether $E(X)$ converges or diverges.

The probability density function of $X$ is $f(x)=kx^{-3}$ for $x\ge1$ and $0$ otherwise. Determine the value of $k$ and state whether $\mathrm{E}(X)$ exists.

Let $X$ have pdf $\displaystyle f(x)=\frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}\quad\text{for }x>0.$ Express $E(X)$ in terms of $\alpha$ and $\beta$.

The pdf of $X$ is defined by $f(x)=\begin{cases} ax, & 0 < x < 1, \\ a(2-x), & 1 < x < 2, \\ 0, & \text{otherwise}. \end{cases}$

Determine $a$ and compute $E(X)$.

Find the expected value of a random variable $X$ that is uniformly distributed on $[0,5]$ with pdf $f(x)=\tfrac{1}{5}$ for $0\le x\le5$.

Find the expected value of a random variable $X$ with pdf $f(x)=4e^{-4x}$ for $x>0$.

Given a random variable $X$ with pdf $f(x)=4e^{-4x}$ for $x>0$, calculate $P(X>1.2)$.