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 $X_{1}, \ldots, X_{n}$ be independent rvs with mean values $\mu_{1}, \ldots, \mu_{n}$ and variances $\sigma_{1}^{2}, \ldots, \sigma_{n}^{2}$ . Consider a function $h\left(x_{1}, \ldots, x_{n}\right),$ and use it to define a new rv $Y=h\left(X_{1}, \ldots, X_{n}\right) .$ Under rather general conditions on the $h$ function, if the $\sigma_{i}$ s are all small relative to the corresponding $\mu_{i} \mathrm{s},$ it can be shown that $E(Y) \approx h\left(\mu_{1}, \ldots, \mu_{n}\right)$ and$\operatorname{Var}(Y) \approx\left(\frac{\partial h}{\partial x_{1}}\right)^{2} \cdot \sigma_{1}^{2}+\cdots+\left(\frac{\partial h}{\partial x_{n}}\right)^{2} \cdot \sigma_{n}^{2}$where each partial derivative is evaluated at $\left(x_{1}, \ldots, x_{n}\right)=\left(\mu_{1}, \ldots, \mu_{n}\right) .$ Suppose three resistorswith resistances $X_{1}, X_{2}, X_{3}$ are connected in parallel across a battery with voltage $X_{4}$ . Then by Ohm's law, the current is $Y=X_{4}\left(\frac{1}{X_{1}}+\frac{1}{X_{2}}+\frac{1}{X_{3}}\right)$Let $\mu_{1}=10 \Omega, \sigma_{1}=1.0 \Omega, \mu_{2}=15 \Omega, \sigma_{2}=1.0 \Omega, \mu_{3}=20 \Omega, \sigma_{3}=1.5 \Omega, \mu_{4}=120 \mathrm{V}$$\sigma_{4}=4.0 \mathrm{V} .$ Calculate the approximate expected value and standard deviation of the current (suggested by "Random Samplings," CHEMTECH, $1984 : 696-697 )$

26, 1.64

Intro Stats / AP Statistics

Chapter 4

Joint Probability Distributions and Their Applications

Section 11

Supplementary Exercises

Probability Topics

The Normal Distribution

Missouri State University

Piedmont College

Oregon State University

Idaho State University

Lectures

0:00

20:37

Let $X_{1}, X_{2},$ and $X…

00:50

Using the notation of Sect…

so he'll have been given that X one extra or Ondo x three are actually the times necessary to perform the three successively bar Roberto us at a certain service off facility on duh We've been given that new one knew too. And Nutri, they are actually the expected values off these random variables. Okay. And similarly Sigma One squared and Sigma Tau Square and Sigma three square These are the villians is off. These expected are, uh, the end of the tables. Now, to start with the first part, it is given to us that a new one is equal to you too is equal to mu tree is equal to 60 and they've also given that Sigma one square is equal. Do Sigmar, it'll square is equal to Sigma three Square and this is equal to 15. So, uh here and what was Hume available by which will be given by the X one less x two less x three So now we're supposed to find the expectation off. Why? So therefore, expectation off by can be organized. Just e off the whole bracket x one plus x to press X tree by using the legality property they can apply expectation to each random variable. So I can drive this as you off x one. Unless I have you off x two, then. Plus we have e off x three. So he off X one is nothing but, um, you want this will be addition off new one last Muto, Last mule tree. So if we had all the body was there all 60 right? So if we add them, then we're getting this value as 1 80 on Deacon all tonight. Expectation off by as you buy next, I'm gonna find the radiance off. Why? So they didn't survive is equal to be off X. One less x. Two less extreme. So now, using the linearity property of radiance we can ride. This is to be off x one. Plus we have the off x two. Plus, next is the off X tree. So you have X one over here. A signal and squares was increasing minds where plastic model square plus Sigma three square. So since sigma one square on de everybody was 15. So we add them to get this as 45. And, uh so Sigma Pi squared is 45. So based on this, we can find out the standard deviation off by which is Sigma by. But you can take the square root off 45 and begin this washer as 6.7 to 8 to moving further. We're supposed to find actually, the probability or for X one less x two plus X three is less than equal to 200. So in terms of if I If I write this, then this is actually called the probability off by less than equal do 200. And now we know the former lovers there it is given by in this case, it'll be by minus. You'd have Mu Bai ah, born sigma. Bye. So then now at the value off, probably over by lessening with 200 is what we're supposed to find So moving further add why equal to 200. I'm going to l Flint. The value off said you're so that will be equal to 200 minus Mu Bai. We're sure is 1 80 divided by Sigma Bias 6.708 toe. So therefore I get the value off their way here as two point nine eq Okay, now so therefore I can say that? Probably Divi less than equal to 200. This actually means that this probability office said there's an equal to 2.98 So this from the table, the center normal table distribution that we have it is equal 2.9986 Then for the second part, they have told us to find probability where 1 50 is less than equal to X. One less extra, less extremely on this is less than equal do 200. So this is what we're supposed to find. Now again, if you know this, we've taken as why already. So then this actually means we're trying to find probability off 1/15 lesson equal. Why is less unequal? Toe 200. So here now, at why equal to 1 50 I want to find the value off said So that is equal to 1 50 minus 1 80 Divided by 6.708 To simplify, we get this as minus 4.47 and for by equal to 200. You already found a value. Sure we have. That is equal to 2.98 So this is from the party, eh? I can therefore write that probability off 1 50 as an equal Y has an equal to 200 in terms of zed, the scan grid and ask probability where this is minus broke. Wind 47 lesson equal does their less than equal to 2.98 So this is actually given by fi off 2.98 minus phi off minus 4.47 So you're going to take these values for understand the normal distribution table. Now, for 2.9 in the value of 0.9986 minus the value of negative 4.47 it is equal to zero. So the answer here getting a 0.9986 again now moving on to the party, They have asked us to find the gravity off. Ex Bart is greater than equal to 55. Now we're supposed to four. Find the expectation ingredients off expert force before we try and find this probability. So move the castle in the expert, which is the mean off all the three variables, so shall be given by X one plus x two plus x three and the whole thing divided by three because we're taking the average shoot. So no, I want to find out the expectation off expert so I can write down that you off X bar is therefore equal to one by three. You can take common and using the geniality property, we can give expectation to each way with this Is he off X one? Unless your fix too. Then I have bless you off x tree. And now this is one by tree. So we have This is new one less muto, less new tree. So this would be equal to 1 80 divided by three. So I don't forget expectation off X box is equal to same U X bar and that is equal to 60. Okay, now moving further. We're also going to find evidence of expert or sure, so for this I'm going to take this as X the variance we will, right? This affects one by three less extra by three unless I have extreme by three. Because while taking the property of variance, the coefficient of the variables are squared. So this changes to one by nine into Sigma one square. Unless we have won by nine into sig model square plus one by nine into Sigma Tree square because the other variances off extra next to an ex tree stand. If I take one bite and I in common on if I add the other values we have this as 45. So then this changes to five. So this is Sigma X Bar Square that we have just calculated. From this, we can find out the standard deviation. So therefore Sigma X bar is equal to the square root of radiance Off expert So which is equal to a squared off five. And this can be taken as 2.2361 Now, next part is you are as but the question we're supposed to find probably off expert created unequal toe 55. So this I can write down as one minus B off ex body is less than 55. So then act X bar equal to 55. We can find out there. Suzanne is equal to expert, which is 55 minus mu X bar, which we've got little as 60 divided by Sigma X bar. So that is 2.2361 So if you simplify, we're getting there for the value of said here as minus 2.4. So then therefore, this probably that we're trying to find in terms of is aided camera and as one minus probability office said is less than minus 2.4. So then this is equal to one minus from the table 2.4 Negative. We can get this at 0.1 through five. And if you subtract down 0.9875 not going on to the next five. Next is what we're supposed to find his probability off X bar between 58 62. So they're supposed to find this now I only know the formula for, said his expert minus new expert upon Sigma X bar. So at expired equal to 58 I'm gonna find all said so, which is equal to 58 minus 60 divided by 2.2361 So I'm getting this that value as minus 0.89 and further at expert Equal toe 62. I wonder if I know said so. This is 62 minus 60 divided by 2.2361 So here I get the value of their again as zero point 89 So therefore, in terms of their that this would be equal to probability. Off minus 0.89 Pleasant Equal does their pleasant equal to 0.89 So this is equal Do fi off wind 89 minus try off minus wind 89 So then, from those Steiner Normal distribution table this first value is zero wind 8133 minus the second value is 0.1867 So if we subtract Dion So we're getting here 0.6266 moving on to the next part party the boulders to find probability where we have my understand less unequal do x one minus wind five x two minus 0.5 x three His lesson equal to five So this is what we're supposed to Fine. So here to start with m wantto take this function your rst So we say let d the equal to x one minus wind five x two minus 0.5 x three So nice just to find the expectations. Thailand Aviation off key. So then I can say therefore, expectation off the can be given By using the banality property, we can say this will be e off x one minus 0.5 into this is E off x two and minus 0.5. And do we have the off X tree? So then this is equal. Do you have? X one is new one which is 60 minus this 0.5 into 60 again minus 600.5 in tow. 60 So I dare forget new duties. He off t is equal to zero. Next is we're supposed to find the variance off so we off b is equal to be off. We have x one minus 10.5 x two minus 0.5 x three Using the property of aliens, the coefficients off the variables will get square So we have you off exponents Sick mom and square. Then we left plus 30.25 Indo Sigma Tau square less point proof Ivindo Sigma three square. So sigma one square. This 15. So we have 15 last 150.2 515 plus 0.25 into 15 city cars like this. Then we're getting the value your as 22.5. So this is Sigma T Square. Therefore the stand Indication Sig Marty squared off the variance. So that is a swear word off 22.5. So if you're simply fire, we're getting this has 4.7434 Yeah. Since we've been told to find more probability of minor stand less than equal dealers and equal to five so said violin sure will be equal to B minus mu t upon Sigma t. So now, at the equal to minus 10 I want to get laid. The value absurd. So this is minus 10 minus zero, divided by Sigma T is full 0.74 34 So if you cast let this then I'm getting the value here as minus 2.11 Next to that be equal to five. That is equal to five minus zero. Born 4.7434 So I get the valley here as 1.5 So therefore, in terms of that, this will be the probability of minus 2.11 less than equal. Who's there? Hasn't equaled 1.5 So which is equal to fi off 1.5 minus phi off minus to find 11 now from the standard normal distribution table we can take these values on. We can substitute them over here. So this value is 0.8531 minus the next value from the table is 0.174 So it's up crack down. So we're getting a 0.83 57 not moving on to the next part for the party E. They have given their Jane the data and they've said that new one is now 40. Then you do is 50. And, um, you tree is 60 then we have signal and squared is thin. Next, Sigmar Total square is 12 and Sigmar three square is 14 And they told us to find ah, the addition of Exxon extracts to use less than equal to 1 60 So I'm going to write down that lead w will be equal to x one less extra plus x tree. So now therefore, to find expectation off w If I apply the inanity property, this will be new one last Muto plus new tree. So if I add all of them 40 plus 50 plus 60 this will give me the value as 1 50 So I've got the value off new W access to find the wedding's off w So again applying the community property, it's really be equal to Sigmund Square Lessig Model Square Less stigma three square. So if you add 10 plus 12 plus 14 we get this valley here as 36. So next is to find the standard deviation Sigma W, which is squared off of aliens 36. So this gives you six. So therefore, to find the probability off, X one plus extra, less extreme is less than equaled 1 60 So we can write this as probability off w less unequal do 1 60 we're sure we can Katherine devalue off there and W quarto 1 60 so said will be equal to W, which is 1 60 minus mute of new is 1 50 divided by Sigmund of Blue is six. So therefore, the value of said that I'm getting here from the table. Oh, from Cal Fielding. This is 1.67 and so therefore probability off they're less than equal to 1.67 is the regular table value at 1.67 So which is equal to 0.955 and the last part further. We're going to assume sure C k is equal to x one plus X to minus two x three. So again, using the same process, we will find out the expectation off K. So using the legality property off expectation, this would be equal to expectation of X one, which is a new one. Less expectation of extra, which is Muto minus two. Expectation of X three, which is Nutri Soapy substance. The values off anyone new to new Tree and Cath late than the value that we will get here from Yuki is equal. Do minus 30. Next, we're gonna find out of aliens off K. So therefore, ratings off Kay will be equal. Do variance off X one less X to minus two. Extremely, it's using the property's off regions. This will change to Sigmund Square, which is the off X one plus Sigma Tau squared is vey off X to minus two will change two plus four and we off extra years Sigma tree square. So substituting the values here we will get this as 78. So therefore the standard division for Kay is squared off the villians and square root off 78 there's a 0.83 one eak. So after this, they told us to calculate the probability off X one bless extrude is greater than equal to twice off x tree. So I can rewrite this part here and say that you off x one plus x two minus twice off X three is greater than equal to zero. So which actually means this is probably off k greater than equal to zero. And this part I can write down as one minus p off key less than zero. So no therefore act okay equal to zero and want to find on the value of that. So that will be equal to zero Minus mu came UK is minus 30. They rendered by Sigma Ki So which is it? Wind it chri 18 So if you simplify, we don't forget the value of their eyes three wind 39 And therefore to find the probability off x one plus X to greater than equal to twice off X tree. This ultimately is equal to one minus B off zed less than 3.39 So from the table, we're goingto take the value of that, which is a 3.3 nights. So which is 0.9997 And if you simply fire supplied the answer, we're getting a 0.3 So we're done with all the five parts off simplifying this particular problem.

View More Answers From This Book

Find Another Textbook

05:39

Bergans of Norway has been making outdoor gear since $1908 .$ The following …

01:10

Consider a regression study involving a dependent variable $y,$ a quantitati…

02:31

The Wall Street Joumal's Shareholder Scoreboard tracks the performance …

02:20

One of the questions on the Business Week Subscriber Study was, "In the…

09:11

The number of individuals arriving at a post office to mail packages during …

04:10

The chamber of commerce of a Florida Gulf Coast community advertises that ar…

05:52

In exercise 18 , the data on grade point average $x$ and monthly salary $y$ …

04:38

In cost estimation, the total cost of a project is the sum of component task…

06:43

The following data were used in a regression study.$\begin{array}{cccccc…

09:12

The following $3 \times 3$ contingency table contains observed frequencies f…