// Numbas version: finer_feedback_settings {"name": "Week 2 quiz", "metadata": {"description": "", "licence": "None specified"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-shuffled", "pickQuestions": 1, "questionNames": ["", "", "", "", "", "", "", "", "", "", ""], "variable_overrides": [[], [], [], [], [], [], [], [], [], [], []], "questions": [{"name": "Efficient mean and variance for random sums of IID random variables", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "adam bretherton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18177/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "

Using generating functions is always the most efficient way to determine the mean and variance of a random sum of iid
random variables?

", "advice": "

Not necessarily, the law of total expectation and variance might be a quicker way to find
these quantities.

The MGF of the standard Normal distribution ($\\mu = 0, \\sigma^2 = 1$) is given by $\\frac{s^2}{2}$?

", "advice": "

This is actually the CGF of the standard normal.

The MGF is $\\exp(\\frac{s^2}{2})$

When does the skewness of the Binomial distribution with paramaters $n, p$ have the following properties?

", "advice": "

The third cumulant of the Binomial distribution is $np(1-p)(1-2p)$ which is:

$0$ when $p=0.5$
Positive when $p<0.5$
Negative when $p > 0.5$

This can be seen by examining the $1-2p$ term, since the $np(1-p)$ term is always positive.

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Which of the following distributions have all their cumulants the same?

", "advice": "

All of the cumulants of the Poisson are equal to its mean.

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For the normal distribution, which of the following is equal to 0?

", "advice": "

This is found by deriving, $m_X(s)$, $M_X(s)$ and $\\log(m_X(s))$ three times.

By inspecting all three of these we see that only the MGF will not bring an $s$ outside the exponential during derivation and so will be the only one of these to not equal zero.

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Discrete random variables do not have an MGF.

", "advice": "

Discrete random variables can have an MGF, and it is not difficult to convert a PGF into an MGF $m_X(s) = P_X(e^s)$.

Of course, continuous random variables cannot have a PGF.

Choose the appropriate option.

", "advice": "

The raw moments are found with the MGF $m_X(s)$

The central moments are found with the MGF about the mean $M_X(s)$

The cumulants are found with the CGF $\\log(m_X(s))$

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Which of these is true about an MGF?

", "advice": "

$m(0) = \\mbox{E}[e^0] = \\mbox{E}[1] = 1$ so the MGF always exists at $s=0$

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Select which option is correct for the formula for obtaining the expected value from the MGF.

", "advice": "

We use $s = 1$ for PGFs but $s = 0$ for MGFs when computing expectation based on evaluating the derivative of the generating function.

For $X_i \\overset{\\text{iid}}{\\sim} X$.

And $N$ as a discrete random variable.

Let $Y = X_1 + X_2 + \\cdots + X_N$.

Which of the following is True?

", "advice": "

Yes we can use the law of total expectation and variance.

The law of total expection is $E[Y] = E[E[Y|N]]$.

The law of total variation is $Var[Y] = E[Var[Y|N]] + Var[E[Y|N]]$.

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All continuous random variables have an MGF?

", "advice": "

The MGF does not exist for some continuous distributions, that is we cannot find an interval around $s = 0$ for which $E[e^{sX}]$ exists. One example is the Cauchy distribution, which does not have any finite moments.

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