IB Mathematics Analysis and Approaches Question Type 7: Finding the Variance for a Given Continuous Distribution and PDF Exercises

Let $X$ have pdf $f(x)=k{x}^3$ for $0<x<1$ and $f(x)=0$ otherwise. Find $\mathrm{Var}(X)$.

Let $X$ have pdf $f(x)=k{x}^{-1/2}$ for $0<x<1$ and $0$ otherwise. Find $\mathrm{Var}(X)$.

A triangular pdf is given by $f(x)=k(2-x)$ for $0\le x\le 2$ and $0$ otherwise. Find $\mathrm{Var}(X)$.

Let $X$ have probability density function $f(x)=k(x-1)$ for $1<x<3$ and $0$ otherwise. Find $\mathrm{Var}(X)$.

For $\lambda >0$, let $X$ have exponential pdf $f(x)=\lambda e^{-\lambda x}$ for $x\ge 0$. Show that $\mathrm{Var}(X)=\frac{1}{{\lambda }^2}$ by direct integration.

Let $X$ have pdf $f(x)=k{x}^2$ for $0<x<2$ and $0$ otherwise. Find $\mathrm{Var}(X)$.

A continuous random variable $X$ has the probability density function $f$ given by $$f(x)=\{\begin{array}{ll}kx,& 0<x<1\\ k(2-x),& 1\le x<2\\ 0,& \text{otherwise}\end{array}$$ Find $\mathrm{Var}(X)$.

Let $X$ have pdf $f(x)=k(4-x^2)$ on $-2\le x\le 2$ and $0$ otherwise. Find $\mathrm{Var}(X)$.

Consider the probability density function $f(x)=\frac{c}{(1+x{)}^3}$ for $x\ge 0$ and $0$ otherwise. Determine $\mathrm{Var}(X)$ (state if it does not exist/ is infinite).

Let $X$ have pdf $f(x)=cx{e}^{-x/2}$ for $x\ge 0$ and $0$ otherwise. Find $\mathrm{Var}(X)$.

Let $X$ have pdf $f(x)=1-|x|$ for $-1\le x\le 1$ and $0$ otherwise. Find $\mathrm{Var}(X)$.

A random variable $X$ has probability density function $f(x)=k\sqrt{2-x}$ for $1\le x\le 2$ and $0$ otherwise. Find $\mathrm{Var}(X)$.