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Random sums. If $X_{1}, X_{2}, \ldots, X_{n}$ are independent rvs, each with the same mean value $\mu$ and variance $\sigma^{2},$ then we have seen that $E\left(X_{1}+X_{2}+\cdots+X_{n}\right)=n \mu$ and $\operatorname{Var}\left(X_{1}+X_{2}+\cdots+X_{n}\right)$$=n \sigma^{2} .$ In some applications, the number of $X_{i}$ s under consideration is not a fixed number $n$ butinstead a rv $N .$ For example, let $N$ be the number of components of a certain type brought into a repair shop on a particular day and let $X_{i}$ represent the repair time for the $i$ th component. Thenthe total repair time is $T_{N}=X_{1}+X_{2}+\cdots+X_{N},$ the sum of a random number of rvs.(a) Suppose that $N$ is independent of the $X_{i}$ s. Use the Law of Total Expectation to obtain anexpression for $E\left(T_{N}\right)$ in terms of $\mu$ and $E(N) .$(b) Use the Law of Total Variance to obtain an expression for $\operatorname{Var}\left(T_{N}\right)$ in terms of $\mu, \sigma^{2}, E(N)$$\quad$ and $\operatorname{Var}(N) .$(c) Customers submit orders for stock purchases at a certain online site according to a Poissonprocess with a rate of 3 per hour. The amount purchased by any particular customer (in thousands of dollars) has an exponential distribution with mean $30 .$ What is theexpected total amount (\$) purchased during a particular 4 -h period, and what is thestandard deviation of this total amount?

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Intro Stats / AP Statistics

Chapter 4

Joint Probability Distributions and Their Applications

Section 1

Jointly Distributed Random Variables

Probability Topics

The Normal Distribution

Missouri State University

Piedmont College

University of St. Thomas

Boston College

Lectures

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You will develop some impo…

Yeah. Um So for the eight but so we have eight. The expectation of S. N. So here the mean will be equal to the expectation of expectation of Sn giving an in. So this will be equal to the expectation of N. You Yes any over here and find the expectation you're going to get me expectation of. And so this is statements and for the first time which is the expectation of X. And all the equal to mean expectation. Okay so for the people that which is the variance of X. N. Okay this is going to be called to the expectation oh the Syrians of S. M. Giving in class Liberian's of expectation. Oh yes and giving in. Yeah. So this is going to be thought. So you're going to get so for the first pass we have expectation of mm sigma that was we have an expectation of any sigma squared plus. So for the variance part of a here we communicate far off any mm So from here you're going to get sigma squared times expectation of in plus we have immune squared far off and which is the variance of. Yeah. So the variance of S. N. Is not equal to sigma squared expectation of N brush being scared. The variance of. And so for the sea baths we have C giving you a present to have a rate of three. So we have a cousin Having a raise of three And amongst purchased which has an exponential distribution with mean of 30. So we have access is financially distributed with a mean of dirty over here. And as we know for a Poisson process that. Nt represent the number of customers ordered for stock purchase. So we know N. T. To be ego too. Then Manitoba. Mhm. Number of customers. Let me see a strict in it. So this way implies that Mt has it wasn distribution with our party. Why? We know alpha to be equal to three. And we know time T. Is equal to four. So this will be equal to So we are going to get empty having a poison of two. So we cannot find them. The expectation of S. N. So we know this to be called expectation of end times meal. You know immune to be called dirty. And we know expectation of N. It's true. So we are going to get Mhm. 30 thanks to Yeah. Peace. So this will give us a 3 60 Dallas overhead. Mhm. So to find for the variance of S. N. Mhm. You know it's sigma squared. An expectation of in plus immune squared the variance of okay, You know sigma is 30. So you have 30 squared. We know expectation of any stool. So thanks. So over here plus you know mean to be equality. So we have taxi squared and we know that far off in which is the experience of in also to be to all. So we are going to get my 100. Thanks 24. Yeah which is equal to 21 Is there is there is zero. So we have 21,000. Oh, we have $210.

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