Question Type 5: Finding the mode for a given PDF Bootcamps

Find the mode of the PDF $f(x)=x^2$ for $0<x<1$.

Find the mode of the PDF $f(x)=1-(x-1)^2$ for $1<x<2$.

Let $f(x)=k\bigl(1-(x-1)^2\bigr)$ for $1<x<2$ and $f(x)=0$ otherwise. (a) Find $k$. (b) Determine the mode of $f$.

For the PDF

$f(x)=\begin{cases} x^2,&0<x<1,\\ 1-(x-1)^2,&1<x<2,\\ 0,&\text{otherwise,} \end{cases}$

determine the intervals on which $f(x)$ is increasing and decreasing, and use this to justify the mode location.

Consider the piecewise PDF

Find the mode(s) of this distribution.

Let $f(x)=k\,x^2$ for $0<x<1$ and $f(x)=0$ otherwise. (a) Determine $k$ to make $f$ a PDF. (b) Find the mode of this distribution.

For the piecewise PDF

derive the cumulative distribution function $F(x)$ for all real $x$, and then identify the mode.

A random variable $X$ has the piecewise PDF

Compute $f'(x)$ on each interval, determine the critical point, and hence find the mode.

Suppose $X$ has PDF $f(x)=k\,x^{\alpha}$ for $0<x<1$, where $\alpha>0$. (a) Determine $k$. (b) Find the mode in terms of $\alpha$. (c) Evaluate the mode for $\alpha=2$.

Given the PDF

$f(x)=\begin{cases} x^2,&0<x<1,\\ 1-(x-1)^2,&1<x<2,\\ 0,&\text{otherwise,} \end{cases}$

compute the second derivative $f''(x)$ at the mode and verify it is a maximum.

Let $Y=X^3$ where $X$ has PDF $f_X(x)=x^2$ for $0<x<1$ and $0$ otherwise. Find the mode of $Y$.

Let $Y=\ln X$ where $X$ has PDF $f_X(x)=x^2$ for $0<x<1$ and $f_X(x)=0$ otherwise. Find the mode of $Y$.