Bootcamp for Question Type 7: Finding the variance for a given continuous distribution and PDF - IB | RevisionDojo

Question Type 7: Finding the variance for a given continuous distribution and PDF Bootcamps

Question 1

Skill question

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

Question 2

Skill question

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

Question 3

Skill question

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

Question 4

Skill question

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

Question 5

Skill question

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

Question 6

Skill question

A random variable $X$ has pdf $f(x)=k\,\sqrt{2-x}$ for $1\le x\le 2$ and $0$ otherwise. Find $\operatorname{Var}(X)$ .

Question 7

Skill question

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

Question 8

Skill question

A piecewise linear pdf is given by

f(x)=\begin{cases} kx, & 0<x<1,\\ k(2-x), & 1\le x<2,\\ 0, & \text{otherwise.} \end{cases}

Find $\operatorname{Var}(X)$ .

Question 9

Skill question

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

Question 10

Skill question

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

Question 11

Skill question

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

Question 12

Skill question

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

Question 1

Skill question

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

Question 2

Skill question

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

Question 3

Skill question

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

Question 4

Skill question

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

Question 5

Skill question

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

Question 6

Skill question

A random variable $X$ has pdf $f(x)=k\,\sqrt{2-x}$ for $1\le x\le 2$ and $0$ otherwise. Find $\operatorname{Var}(X)$ .

Question 7

Skill question

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

Question 8

Skill question

A piecewise linear pdf is given by

f(x)=\begin{cases} kx, & 0<x<1,\\ k(2-x), & 1\le x<2,\\ 0, & \text{otherwise.} \end{cases}

Find $\operatorname{Var}(X)$ .

Question 9

Skill question

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

Question 10

Skill question

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

Question 11

Skill question

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

Question 12

Skill question

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