Moments

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Problems

Exercises marked with s have detailed solutions at http://stat110.net.

Means, Medians, Modes, and Moments

6.1

Let $U\sim \mathrm{Unif}(a,b)$. Find the median and mode of $U$.

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6.2

Let $X\sim \mathrm{Expo}(\lambda )$. Find the median and mode of $X$.

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6.3

Let $X$ have the Pareto distribution with parameter $a>0$; this means that $X$ has PDF

$$f(x)=\frac{a}{x^{a+1}}$$

for $x\ge 1$ (and $0$ otherwise). Find the median and mode of $X$.

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6.4

Let $X\sim \mathrm{Bin}(n,p)$.

(a) For $n=5$, $p=1/3$, find all medians and all modes of $X$. How do they compare to the mean?

(b) For $n=6$, $p=1/3$, find all medians and all modes of $X$. How do they compare to the mean?

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6.5

Let $X$ be Discrete Uniform on $1,2,\dots ,n$. Find all medians and all modes of $X$ (your answer can depend on whether $n$ is even or odd).

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6.6

Suppose that we have data giving the amount of rainfall in a city each day in a certain year. We want useful, informative summaries of how rainy the city was that year. On the majority of days in that year, it did not rain at all in the city. Discuss and compare the following six summaries: the mean, median, and mode of the rainfall on a randomly chosen day from that year, and the mean, median, and mode of the rainfall on a randomly chosen rainy day from that year (where by “rainy day” we mean that it did rain that day in the city).

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6.7

Let $a$ and $b$ be positive constants. The Beta distribution with parameters $a$ and $b$, which we introduce in detail in Chapter 8, has PDF proportional to $x^{a-1}(1-x{)}^{b-1}$ for $0<x<1$ (and the PDF is $0$ outside of this range). Show that for $a>1$, $b>1$, the mode of the distribution is $(a-1)/(a+b-2)$.

Hint: Take the log of the PDF first (note that this does not affect where the maximum is achieved).

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6.8

Find the median of the Beta distribution with parameters $a=3$ and $b=1$ (see the previous problem for information about the Beta distribution).

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6.9

Let $Y$ be Log-Normal with parameters $\mu$ and ${\sigma }^2$. So $Y=e^X$ with $X\sim \mathcal{N}(\mu ,{\sigma }^2)$. Three students are discussing the median and the mode of $Y$. Evaluate and explain whether or not each of the following arguments is correct.

(a) Student A: The median of $Y$ is $e^{\mu }$ because the median of $X$ is $\mu$ and the exponential function is continuous and strictly increasing, so the event $Y\le e^{\mu }$ is the same as the event $X\le \mu$.

(b) Student B: The mode of $Y$ is $e^{\mu }$ because the mode of $X$ is $\mu$, which corresponds to $e^{\mu }$ for $Y$ since $Y=e^X$.

(c) Student C: The mode of $Y$ is $\mu$ because the mode of $X$ is $\mu$ and the exponential function is continuous and strictly increasing, so maximizing the PDF of $X$ is equivalent to maximizing the PDF of $Y=e^X$.

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6.10

A distribution is called symmetric unimodal if it is symmetric (about some point) and has a unique mode. For example, any Normal distribution is symmetric unimodal. Let $X$ have a continuous symmetric unimodal distribution for which the mean exists. Show that the mean, median, and mode of $X$ are all equal.

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6.11

Let $X_1,\dots ,X_n$ be i.i.d. r.v.s with mean $\mu$, variance ${\sigma }^2$, and skewness $\gamma$.

(a) Standardize the $X_j$ by letting

$$Z_j=\frac{X_j-\mu }{\sigma }.$$

Let ${\bar{X}}_n$ and ${\bar{Z}}_n$ be the sample means of the $X_j$ and $Z_j$, respectively. Show that $Z_j$ has the same skewness as $X_j$, and ${\bar{Z}}_n$ has the same skewness as ${\bar{X}}_n$.

(b) Show that the skewness of the sample mean ${\bar{X}}_n$ is $\gamma /\sqrt{n}$. You can use the fact, shown in Chapter 7, that if $X$ and $Y$ are independent then $E(XY)=E(X)E(Y)$.

Hint: By (a), we can assume $\mu =0$ and ${\sigma }^2=1$ without loss of generality; if the $X_j$ are not standardized initially, then we can standardize them. If $(X_1+X_2+\cdots +X_n{)}^3$ is expanded out, there are 3 types of terms: terms such as $X_1^3$, terms such as $3X_1^2{X}_2$, and terms such as $6X_1{X}_2{X}_3$.

(c) What does the result of (b) say about the distribution of ${\bar{X}}_n$ when $n$ is large?

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6.12

Let $c$ be the speed of light in a vacuum. Suppose that $c$ is unknown, and scientists wish to estimate it. But even more so than that, they wish to estimate $c^2$, for use in the famous equation $E=m{c}^2$.

Through careful experiments, they obtain i.i.d. measurements $X_1,\dots ,X_n\sim \mathcal{N}(c,{\sigma }^2)$. Using these data, there are various possible ways to estimate $c^2$. Two natural ways are: (1) estimate $c$ using the average of the $X_j$‘s and then square the estimated $c$, and (2) average the $X_j^2$‘s. So let

$${\bar{X}}_n=\frac{1}{n}\sum _{j=1}^n{X}_j,$$

and consider the two estimators

$$T_1={\bar{X}}_n^2\text{and}{T}_2=\frac{1}{n}\sum _{j=1}^n{X}_j^2.$$

Note that $T_1$ is the square of the first sample moment and $T_2$ is the second sample moment.

(a) Find $P(T_1<T_2)$.

Hint: Start by comparing ${\left(\frac{1}{n}{\sum }_{j=1}^n{x}_j\right)}^2$ and $\frac{1}{n}{\sum }_{j=1}^n{x}_j^2$ when $x_1,\dots ,x_n$ are numbers, by considering a discrete r.v. whose possible values are $x_1,\dots ,x_n$.

(b) When an r.v. $T$ is used to estimate an unknown parameter $\theta$, the bias of the estimator $T$ is defined to be $E(T)-\theta$. Find the bias of $T_1$ and the bias of $T_2$.

Hint: First find the distribution of ${\bar{X}}_n$. In general, for finding $E(Y^2)$ for an r.v. $Y$, it is often useful to write it as $E(Y^2)=\mathrm{Var}(Y)+(EY{)}^2$.

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Moment Generating Functions

6.13

Stat110 solution available.

A fair die is rolled twice, with outcomes $X$ for the first roll and $Y$ for the second roll. Find the moment generating function $M_{X+Y}(t)$ of $X+Y$ (your answer should be a function of $t$ and can contain unsimplified finite sums).

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6.14

Stat110 solution available.

Let $U_1,U_2,\dots ,U_{60}$ be i.i.d. $\mathrm{Unif}(0,1)$ and $X=U_1+U_2+\cdots +U_{60}$. Find the MGF of $X$.

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6.15

Let $W=X^2+Y^2$, with $X,Y$ i.i.d. $\mathcal{N}(0,1)$. The MGF of $X^2$ turns out to be $(1-2t{)}^{-1/2}$ for $t<1/2$ (you can assume this).

(a) Find the MGF of $W$.

(b) What famous distribution that we have studied so far does $W$ follow (be sure to state the parameters in addition to the name)? In fact, the distribution of $W$ is also a special case of two more famous distributions that we will study in later chapters!

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6.16

Let $X\sim \mathrm{Expo}(\lambda )$. Find the skewness of $X$, and explain why it is positive and why it does not depend on $\lambda$.

Hint: Recall that $\lambda X\sim \mathrm{Expo}(1)$ and the $n$th moment of an $\mathrm{Expo}(1)$ r.v. is $n!$ for all $n$.

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6.17

Let $X_1,\dots ,X_n$ be i.i.d. with mean $\mu$, variance ${\sigma }^2$, and MGF $M$. Let

$$Z_n=\sqrt{n}\left(\frac{{\bar{X}}_n-\mu }{\sigma }\right).$$

(a) Show that $Z_n$ is a standardized quantity, i.e., it has mean $0$ and variance $1$.

(b) Find the MGF of $Z_n$ in terms of $M$, the MGF of each $X_j$.

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6.18

Use the MGF of the $\mathrm{Geom}(p)$ distribution to give another proof that the mean of this distribution is $q/p$ and the variance is $q/p^2$, with $q=1-p$.

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6.19

Use MGFs to determine whether $X+2Y$ is Poisson if $X$ and $Y$ are i.i.d. $\mathrm{Pois}(\lambda )$.

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6.20

Stat110 solution available.

Let $X\sim \mathrm{Pois}(\lambda )$, and let $M(t)$ be the MGF of $X$. The cumulant generating function is defined to be $g(t)=\log M(t)$. Expanding $g(t)$ as a Taylor series

$$g(t)=\sum _{j=1}^{\infty }\frac{c_j}{j!}{t}^j$$

(the sum starts at $j=1$ because $g(0)=0$), the coefficient $c_j$ is called the $j$th cumulant of $X$. Find the $j$th cumulant of $X$, for all $j\ge 1$.

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6.21

Stat110 solution available.

Let $X_n\sim \mathrm{Bin}(n,p_n)$ for all $n\ge 1$, where $n{p}_n$ is a constant $\lambda >0$ for all $n$ (so $p_n=\lambda /n$). Let $X\sim \mathrm{Pois}(\lambda )$. Show that the MGF of $X_n$ converges to the MGF of $X$ (this gives another way to see that the $\mathrm{Bin}(n,p)$ distribution can be well-approximated by the $\mathrm{Pois}(\lambda )$ when $n$ is large, $p$ is small, and $\lambda =np$ is moderate).

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6.22

Consider a setting where a Poisson approximation should work well: let $A_1,\dots ,A_n$ be independent, rare events, with $n$ large and $p_j=P(A_j)$ small for all $j$. Let

$$X=I(A_1)+\cdots +I(A_n)$$

count how many of the rare events occur, and let $\lambda =E(X)$.

(a) Find the MGF of $X$.

(b) If the approximation $1+x\approx e^x$ (this is a good approximation when $x$ is very close to $0$ but terrible when $x$ is not close to $0$) is used to write each factor in the MGF of $X$ as $e$ to a power, what happens to the MGF? Explain why the result makes sense intuitively.

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6.23

Let $U_1,U_2$ be i.i.d. $\mathrm{Unif}(0,1)$. Example 8.2.5 in Chapter 8 shows that $U_1+U_2$ has a Triangle distribution, with PDF given by

$$f(t)=\{\begin{array}{ll}t& \text{for }0<t\le 1,\\ 2-t& \text{for }1<t<2.\end{array}$$

The method in Example 8.2.5 is useful but it often leads to difficult integrals, so having alternative methods is important. Show that $U_1+U_2$ has a Triangle distribution by showing that they have the same MGF.

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6.24

Let $X$ and $Y$ be i.i.d. $\mathrm{Expo}(1)$, and $L=X-Y$. The Laplace distribution has PDF

$$f(x)=\frac{1}{2}{e}^{-|x|}$$

for all real $x$. Use MGFs to show that the distribution of $L$ is Laplace.

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6.25

Let $Z\sim \mathcal{N}(0,1)$, and $Y=|Z|$. So $Y$ has the Folded Normal distribution, discussed in Example 5.4.7. Find two expressions for the MGF of $Y$ as unsimplified integrals: one integral based on the PDF of $Y$, and one based on the PDF of $Z$.

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6.26

Let $X,Y,Z,W\sim \mathcal{N}(0,1)$ be i.i.d.

(a) Find an expression for $E\left(\Phi (Z)e^Z\right)$ as an unsimplified integral.

(b) Find $E(\Phi (Z))$ and $E(e^Z)$ as fully simplified numbers.