Workshop questions

Define the following, in the context of continuous random variables.

1. (a)

   Cumulative distribution function

2. (b)

   Probability density function

3. (c)

   Expected value

4. (d)

   Variance

5. (e)

   Quantile

Let $X$ be a continuous random variable with induced sample space $[-3,3]$, and probability density function (pdf) $f_X$ given by:

1. (a)

   Find the cumulative distribution function (cdf) $F_X$.

2. (b)

   Find , $P(-2\le X\le 1)$ and $P(X>2.5)$.

Let $Y$ be a continuous random variable with induced sample space $[0,1]$. Assume $Y$ has probability density function (pdf) proportional to $y(1-y^2)$ on that interval, and 0 elsewhere, i.e. there exists a constant $c$ such that

1. (a)

   Find the value $c$ must take for this to be a pdf.

2. (b)

   Sketch the pdf.

3. (c)

   Find and sketch the cumulative distribution function $F_Y$.

A dart is thrown at a circular target of radios $a$ and always hits some point of the target. The probability the dart hits any particular region of the target is proportional to the area of that region. Let $R$ be the distance between the target centre and the point the dart hits. Find and sketch the cumulative distribution function $F_R(r)$ (it may help to consider $P(R\le r)$ separately for the three cases , $0\le r\le a$ and $r>a$). Hence find and sketch the probability density function $f_R(r)$. Use $f_R$ to find $E[R]$.

Let $X$ be an $\mathrm{Exp}(\lambda )$ random variable with $F_X(x)=1-\exp (-\lambda x)$ for $x>0$ and $F_X(x)=0$ otherwise, and let $Y=cX$ where $c>0$ is a fixed constant. Find the cumulative distribution function of $Y$. By differentiation, find the probability density function of $Y$. What is the distribution of $Y$?