Workshop questions

Define what is meant by the following.

1. (a)

   Random variable

2. (b)

   Induced sample space

3. (c)

   Probability mass function

4. (d)

   Cumulative distribution function

5. (e)

   Expected value

6. (f)

   Variance

7. (g)

   Standard deviation

The rv $R$ has pmf $p_R(r)=1/8$ for $r=1,2,\mathrm{\dots },8$. Find

1. (a)

   $\mathrm{P}(R>2)$,

2. (b)

   The cdf of $R$.

A random variable $R$ has pmf $p_R(r)$, for $r=0,1,\mathrm{\dots }$. Prove the following results concerning the expectation

1. (a)

   $\mathrm{E}[g(R)+h(R)]=\mathrm{E}[g(R)]+\mathrm{E}[h(R)]$,

2. (b)

   $\mathrm{E}[cg(R)]=c\mathrm{E}[g(R)]$,

where $g,h$ are functions and $c$ is constant.

A survey has shown that $20\%$ of pre-school children in the UK have speech or hearing difficulties.

Write down a formula for $p(x)=\mathrm{P}(X=x)$, when $X$ is the number of children in a nursery class of $20$ children who have speech or hearing difficulties.

Write down the expected number of children in the class with difficulties.

Staff claim that, if the class were to contain more than two children with speech or hearing difficulties, then the work of the class could be adversely affected; find the probability of this event.

If $R$ is a geometric random variable, find $\mathrm{P}(R\ge n)$.

Show that

Give a verbal argument as to why this property should be true.