Home page for accesible maths Math230, 2017-18: Workshop, Coursework & Quiz Questions MATH230 Week 01 - Assessed problems (coursework) MATH230 Week 02 - Assessed problems (coursework)

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MATH230 Week 02 - Workshop problems

Please have a go at the problems before the workshop so that in the workshop you can focus on those problems which you had trouble with.

W02.1 Discrete cdf moments

A discrete random variable, $R$ , has a cdf of

\displaystyle F_{R}(r)=\left\{\begin{array}[]{ll}0&\quad r<0\\ 1/9&\quad 0\leq r<1\\ 5/9&\quad 1\leq r<2\\ 1&\quad 2\leq r\end{array}\right.

Use the pmf that you calculated in WS01 Discrete cdf to answer the following.

(a)

What is $\operatorname{\mathsf{E}}\left[{R^{a}}\right]$ for any $a>0$ ?
(b)

What is ${\operatorname{\mathsf{Var}}}\left[{R}\right]$ ?
(c)

What is the skewness of $R$ ?

W02.2 pdfcst moments

Find the expectation, variance, and skewness for the random variable $X$ in the question in WS1 pdfcst.

W02.3 Poisson accidents.

Suppose that the number of accidents occurring on a highway each day is a Poisson random variable with expected number (i.e. $\lambda$ parameter) $3$ .

(a)

Write the pmf of the Poisson distribution with this parameter value.
(b)

Find the probability that 3 or more accidents will occur today.
(c)

Using conditional probability, repeat part (b) if you know that at least one accident occurred today.

W02.4 Indicator function.

Let $A$ be an interval on the real line. For a set $A$ the function $I_{A}(x)$ of $x$ is defined as

\displaystyle I_{A}(x)=\left\{\begin{array}[]{ll}1&\quad x\in A\\ 0&\quad x\notin A\end{array}\right.

For the continuous random variable $X$ , find $\operatorname{\mathsf{E}}\left[{I_{A}(X)}\right]$ . Hint: either consider the distribution of $Y=I_{A}(X)$ and use the known results for that or first consider an interval, $A=[a,b]$ , and draw the picture.