MATH230 Week 01 - Assessed problems (coursework)

Obtain the cdf of the following discrete random variables

1. (i)

   $R$ where for some parameter, $\theta \in (0,1)$,

   1. $𝖯(R=0)={\theta }^2$,

   2. $𝖯(R=1)=\theta (1-\theta )$,

   3. $𝖯(R=2)=\theta (1-\theta )$,

   4. $𝖯(R=3)={(1-\theta )}^2$ ,

   and for $r\notin \{0,1,2,3\}$, $𝖯(R=r)=0$.

2. (ii)

   $R\sim \mathrm{𝖴𝗇𝗂𝖿}(0,m)$.

[marks: 4]

Hint: The discrete uniform pmf is given near the start of Ch3 of your Math230 notes and in your first year notes. Study Figure 2.1 in your notes. Your function should work for any real number $r\in ℝ$. First calculate $F$ at the integers, being careful to distinguish the index of summation from the upper limit of summation. Secondly extend from the integers to the real line, either by stating ranges or using the int function, where relevant.

The lifetime, $X$, in days of a species of butterfly has a density of

1. (i)

   Find the cdf of $X$ in terms of $b$, being careful to show your working.

2. (ii)

   Write down the value of $b$ and explain how you deduced this.

3. (iii)

   Find the probability that (i) a butterfly lives for more than two days, (ii) a butterfly lives for more than half a day.

4. (iv)

   Find the age which only $1\%$ of butterflies reach.

[marks: 7]

A discrete random variable, $R$, has a parameter $\theta \in (0,1)$ and a pmf of

1. (i)

   Use the solution to WS01 Discrete pmf to cdf to write down the survivor function, $S_R(r)$, of $R$.

2. (ii)

   Hence, or otherwise, find $𝖯(R>a+b|R>a)$ for integer $a\ge 0$ and real number $b\ge 0$.

3. (iii)

   What is unusual about the formula for $𝖯(R>a+b|R>a)$?

[marks: 4]

A random variable $X$ has the following cdf.

By considering ${lim}_{i\to \mathrm{\infty }}{F}_X(x)-F_X(x-1/i)$, or otherwise, describe the random variable itself.

[marks: 5]