IB Mathematics Analysis and Approaches Question Type 5: Finding the Mode for a Given PDF Exercises

For the piecewise PDF $$f(x)=\{\begin{array}{ll}{x}^2,& 0<x<1,\\ 1-(x-1{)}^2,& 1<x<2,\\ 0,& \text{otherwise,}\end{array}$$ derive the cumulative distribution function $F(x)$ for all real $x$, and then identify the mode.

Find the mode of the probability density function $f(x)=3x^2$ for $0\le x\le 1$.

Find the mode of the probability density function $f(x)=\frac{3}{2}(1-(x-1{)}^2)$ for $1\le x\le 2$.

A random variable $X$ has the piecewise PDF $$f(x)=\{\begin{array}{ll}{x}^2,& 0<x<1,\\ 1-(x-1{)}^2,& 1<x<2,\\ 0,& \text{otherwise.}\end{array}$$ Compute $f^{\prime }(x)$ on each interval, determine the critical point, and hence find the mode.

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

Find the mode(s) of this distribution.

For the PDF

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

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

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