Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

Let $X_{1}, X_{2}, \ldots$ be a sequence of independent, but not necessarily identically distributed random variables, and let $T=X_{1}+\cdots+X_{n} .$ Lyapunov's Theorem states that the distribution of the standardized variable $\left(T-\mu_{T}\right) / \sigma_{T}$ converges to a $N(0,1)$ distribution as $n \rightarrow \infty,$ provided that $\lim _{n \rightarrow \infty} \frac{\sum_{i=1}^{n} E\left(\left|X_{i}-\mu_{i}\right|^{3}\right {\sigma_{T}^{3}}=0$where $\mu_{i}=E\left(X_{i}\right) .$ This limit is sometimes referred to as the Lyapunov condition forconvergence.(a) Assuming $E\left(X_{i}\right)=\mu_{i}$ and $\operatorname{Var}\left(X_{i}\right)=\sigma_{i}^{2},$ write expressions for $\mu_{T}$ and $\sigma_{T}$(b) Show that the Lyapunov condition is automatically met when the $X_{i}$ s are iid. [Hint: Let$\tau=E\left(X_{i}-\mu_{i}\right\}^{3} ),$ which we assume is finite, and observe that $\tau$ is the same for every $X_{i}$ . Then simplify the limit.(c) Let $X_{1}, X_{2}, \ldots$ be independent random variables, with $X_{i}$ having an exponential distribution with mean $i .$ Show that $X_{1}+\cdots+X_{n}$ has an approximately normal distribution as $n$ increases.(d) An online trivia game presents progressively harder questions to players; specifically, theprobability of answering the $i$ th question correctly is 1$/ i .$ Assume any player's successiveanswers are independent, and let $T$ denote the number of questions a player has right out ofthe first $n .$ Show that $T$ has an approximately normal distribution for large $n .$

Intro Stats / AP Statistics

Chapter 4

Joint Probability Distributions and Their Applications

Section 11

Supplementary Exercises

Probability Topics

The Normal Distribution

Missouri State University

Cairn University

Idaho State University

Boston College

Lectures

0:00

27:31

Let $X_{1}, \ldots, X_{n}$…

11:11

Let $X(t)$ be a WSS random…

20:15

The article "Statisti…

07:53

Baseball In the year $2014…

15:59

Imprints Galore buys T-shi…

02:45

One way to solve a probabi…

05:41

Let $X$ be a nonnegative c…

07:23

Normal distribution An imp…

03:12

20:37

Let $X_{1}, X_{2},$ and $X…

02:39

06:22

Karl Pearson once tossed a…

03:36

Let $X$ and $Y$ be the tim…

26:45

09:37

A large but sparsely popul…

04:25

For each random variable d…

12:32

Did you ever buy an incand…

14:27

Refer back to the previous…

in problem. 10 were given that there's a random sample X one through xn with a distribution with the mean mu and a variance Sigma scribe. We're going to use this information to to derive set in relationships, questions. And so in the first part of the question A we will be showing that this submission off X. I mean, it's ex boss squad equals the submission of X Grad I minus n x box crab. So illustrate that expression that equation And then we show how it's going to be the ranked so a it's going this submission off X I minus X, uh, squared. He's same us. The submission off X I square minus and x box Quit. Right? So we need to solve the left hand side of the equation, too, so that we can get the right hand side of the questions. So let's begin the expression X I minus X ba is being squared. So when we open up the brackets is going to be the submission off x I minus X box multiplied by itself next, I mean aspects bar, which will be equal to submission both when you open up when you solved this brackets, it will be x Hi multiplied by X I that's going to mean X bisque red then X I was played by X bob would be X I exper next we'll have negative X bar multiplied by X I and that will also be negative. X i ex bum and lasting negative X bomb was played by negative expert will be positive X bar square Now we can simplify this in to be this submission Both x I squared minus capote Stoute electives together to x I X box plus x bar squared. Now that we have got into that point, we can uh huh open up the brackets to show that each of these values would be some TV, some each of this. Then we add them up. It will be the same thing. So it's going to be when we open up. Thats practices will be the submission off X I squared minus the submission both two x i x bar plus the submission move X bar And since in this case we know that the sample sizes And according to this statement, we know that the sample sizes and so this will become the submission this will remain a submission off X. I squared because the values changing minus We could factor out the to. We can remove the two from the submission because it's a constant. So it's going to be, too. And we can also removed the X Bar because it's a constant. So it's been to be expired and then we'll be left with submission off X I plus submission off X bar script. Now, uh, we know that when we sum up when we some I will get the total value off I when you some X I we're going to get a total value of X, and that means we can find a way of replacing these and put the X box. So remember that X box he's obtained from the some off X I divided by n So once we have, we can multiply when you multiply both sides by n it will be in X bar equals the submission off X I and that's what we're going to substitute here next we we know that this is a constant that his ex bar squared is a constant. And so what we need to do in this case is too. Summit end times. So when you some end times, we'll get in X Bhaskaran. So let's move on to the next line. And so this was going to be the submission off X I scribe minus two x box multiplied by in Expo, which we have seen here because it same as the submission of X I plus this some off ex bass squared end times will be in X bar square, not you. Move on and you can move on to the next part which will be simplifying this plant which will give us the submission off X I squared minus two x box multiplied by an ex bomb will become true end X bar squared plus in X box. Quit No, This too can be would be combined so that when we have minus two and expert plus an expert, the results will be negative and experts So this simply is simplified to us. The submission off x I it's quiet minus and x bob squid and this is what we had originally so confirms our proof that these two equations these two expressions are the same. Next impact be off. The question would be showing that Uh, okay. Puppy will be showing that the expectation off the submission off X I squared equals and Miss Crab Plus Signal Square. So we need to work out the expectation off the submission off X eyes squared and see whether it's going to give us the the left hand side of the equation. So let's begin on. We're going to use the the linearity of expectation, along with the fact that the expectation off y squared I need to know that expectation of y squared equals the variance. Why, plus expectation, the expectation off. Why square? Okay, so let's move on and start solving there right hand side of the equation. So the expectation off the submission Off X Ice Grad would be expressed as follows. It's going to be expectation off X one squared, plus all the values until we get to this on the last Monday in the simple That's Ex N Squared. Then this will be separated because the expectation is lenient. So it's going to be expectation off X one squared plast expectation of next one x two squared all the way till we get to the expectation off X n scratch and using this our relation, we'll see that the result here will be the variance off. Why the violence of excess? Sorry, plus the expectation over X squared. But we know of expectation of X. According to this distribution, expectation of X is the mean. So it's going to be plus mu square. Then we add all through for all values of X and two because this is only four x one. So we're going to have all the values of end for the accented gets to the last one. So it's going to be the variance off Ex n plus the mean square. Now we can go ahead to simplify this the variance off X one. According to this distribution, the variance is Sigma squared. So we're going to use sickness crowd for this Cigna squared plus muse squared home to be until they get to the last datum sickness crowd plus new square. And remember, we're going to be summing end types. So this means that at the end we would have in multiplied by Sigma squad from US mu script and we have succeeded in obtaining the final expression which is equal to what we have on the right hand side. Next, we're going to use the same Bob distribution to show but see, going to be showing that the expectation off and X bar squared equals and Meuse crab plus Sigma scrap. And here we're going to apply the relation given in the previous heat. And this is the relation that we were given. So this is what we're going to use in this case just at this time. We replace the value off why with X bar. So that means it's going to be the expectation off Expo equals the variance of Y Palm. Sorry off Expo, plus the expectation Wolf X bar. It's quit. So this is what we're going to use for this problem. Okay, so let's begin looking at the left hand side of expression the Christian. And this is what we see. The expectation before you can start getting the expectation. Let's solve the the value off X bar scrap. We want to simplify X MMA scrap. So, uh, and since we already know that there's a relationship here, it's going to be a expectation off in X bar screen. Since we we have a constant here on the constant is simple. And so we can, we can move it out of the expectation and we'll have n expectation off experts grant. Good. So expectation off ex bass squid This case on the scribe on in this case, this quays Yeah, and express. So in this case, what we need to do is to rewrite it has fallen using this relationship here, it's going to be the expectation it's going to be n was played by the variance off ex bomb plus the expectation off Expo Square. Good. Now we can move on to walk out because we know from the distribution Uh, the variance off X bar would be the variance of expert will be Sigma Squared over n We have n right next to it, plus the expectation off, uh x bar. He's mu. But this time it's going to be squared. And the reason why this end comes up here is because the expression years, all of this we just need to multiply end by all this movie. It's important Twitter record here. Now we can simplify this and we will look Team Sigma's credit plus in new scrap. And so we see that expectation off n expats crowd well, results into Sigma Squared Plus and Meus crab, which is the same was expression that we have here the one that we started with from the beginning. Good. So the party of the question will be combining what we have seen from A to C, uh, to show that s crowd is on unbiased estimate er off Sigma squared. So to show, that s Gradison and vast estimate off sickness crab. This is what we need to show. So we need to show that expectation off s grad equals signal. And we need to use everything that we have seen from the previous part of the question that with previous parts of the question So the fasting way saw the part A in part, Jerry So that the submission off x i minus X box squared becomes the submission of X Y squared minus end ex bass grant. And that's what we're going to use fast. And we're going to give the expression from the, uh, with expectation off s so let's begin. So the expectation of s squared simply means that we're going to be getting the expectation off the following expression one off N minus one. This submission off x I minus X box scratch. Next, we can rewrite this expression. As for us, so we can rewrite is as follows. We can see this is equal to the expectation off one over n minus one On Instead of writing the submission off X I menace, experts squared. We go back to party and we can write it in this form. You can write it as the submission off X I squared minus X box grab. So the submission x I squint minus in next body script, I could and at this point we can remove the constant one of our 10 minutes one from the expectation so that we're left with full length. It's going to be one over n minus one. Would you plant that expectation off this admission of X I squared minus n x square. Good. We noticed that from part B become obtain a different expression for the submission off X I square. So was concealing Good to be. It's the expectation off The submission of X I squared is given by N and is new squared plus sigma squared and that's what we're going to do here. Guntram substituted here end for en ex bass squared. We'll use what we got from Patsy, which is N news grad plus Sigma's crab. So So let's move on. Get these expectations So it's going to be equal to one over n minus one. And we have seen that expectation Wolf X I squared will be and news grad plus sickness crab according to park Be and expectation off and, um, ex boss squared will be and music rat minus sigma squared. Now we can simplify this as full us. So it's going to be one over n minus one. Then you have the bracket there and this will be n sigma squared when you open up class. And so the enemy's got lost and sickness crowd minus and muse clad plus sigma squared. So we have two lecterns which we're going to council not we have an you scratch and discuss with the cancel out and this is what we get. So we have one over ending this one. What is left his end sickness clad plus sigma squared. Now notice that these two terms both have sickness Grab so we can fucked ate out and this would be one of n minus one. Then we fucked out the Cygnus craft. And when we do that, we'll be having in that's one cake. So one thing that we need to do I'm sorry we have to change something here. Going back to what? See Patsy, give us. Plus, here it's correct. This this is supposed to be plus and since this is plus, then this will be minus. So that's changed this to minus. Then this also will be minus. And this will p minus good. And now that we have n minus one in both, um, the new monitor here and the denominator become council they're not. So when you counsel this heart this to you will remain with Sigma Scrap. So the expectation off esque squared is Sigma spread and that confirms that esque read one of the population. The sample variance is an unbiased estimator off the population variance and lastly, in parts E will begin to be telling whether the sample standard Division s he's on unbiased estimator off mu script. Another once whether expectation off yes will be equal to oh, that stigma. Now, remember that for us to move from a variance to the standard deviation, we have to get this crab it. And as we move from the population variance to the the population, standard deviation will also have to get this squared. So in in that case, we would conclude that this is not the case. Expectations off s is not equal to their populations. Standard deviation. Because, as we saw Elliot, expectation, function expectation is Kenya. But the square root function is nonlinear. So as we get the scratch it it's going to be non linear. And for that reason we can conclude that the sample standard deviation is not on and best estimate er off the population standard deviation.

View More Answers From This Book

Find Another Textbook

08:17

Suppose two identical components are connected in parallel, so the system co…

07:48

Refer to the data presented in exercise $2 .$ The estimated regression equat…

03:38

The data in the following table show the number of shares selling (millions)…

23:27

The joint cumulative distribution function of two random variables $X$ and $…

01:43

Recall that in exercise $44,$ the admissions officer for Clearwater College …

07:32

A circular sampling region with radius $X$ is chosen by a biologist, where $…

10:26

$\begin{array}{l}{\text { Small cars offer higher fuel efficiency, are easy …

06:45

One of the biggest changes in higher education in recent years has been the …

05:04

$$\begin{array}{c}{\text { Refer to exercise } 5 .} \\ {\text { a. Use } \al…

05:43

Sporty cars are designed to provide better handling, acceleration, and a mor…