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Question 1

For the piecewise PDF

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

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

[6]

Question 2

Let $f(x)=k\,x^2$ for $0<x<1$ and $f(x)=0$ otherwise.

1.

(a) Determine $k$ to make $f$ a PDF.

[3]

2.

(b) Find the mode of this distribution.

[2]

Question 3

Let $f(x)=k\bigl(1-(x-1)^2\bigr)$ for $1 < x < 2$ and $f(x)=0$ otherwise.

1.

Find the value of $k$ .

[3]

2.

Determine the mode of $f$ .

[2]

Question 4

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

[3]

Question 5

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

[3]

Question 6

A random variable $X$ has the piecewise PDF

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

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

[5]

Question 7

Suppose $X$ has probability density function (PDF) $f(x)=k x^{\alpha}$ for $0<x<1$ , where $\alpha>0$ .

1.

Determine the value of $k$ .

[2]

2.

Find the mode of $X$ in terms of $\alpha$ .

[3]

3.

Evaluate the mode for $\alpha=2$ .

[1]

Question 8

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 $Y$ .

[5]

Question 9

Consider the piecewise PDF

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

Find the mode(s) of this distribution.

[4]

Question 10

For the PDF

f(x)=\begin{cases} x^2,&0 \leq x \leq 1,\\ 1-(x-1)^2,&1 < x \leq 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.

[4]

Question 11

Given the probability density function (PDF): $f(x)=\begin{cases} x^2, & 0 < x \le 1 \\ 1-(x-1)^2, & 1 < x < 2 \\ 0, & \text{otherwise} \end{cases}$

1.

State the mode of the distribution.

[1]

2.

Find $f''(x)$ for the interval $1 < x < 2$ .

[2]

3.

Justify that the mode is a maximum.

[2]

Question 12

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

[3]

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