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A circular sampling region with radius $X$ is chosen by a biologist, where $X$ has an exponentialdistribution with mean value 10 ft. Plants of a certain type occur in this region according to a(spatial) Poisson process with "rate"..5 plant per square foot. Let $Y$ denote the number of plantsin the region.(a) Find $E(Y | X=x)$ and $\operatorname{Var}(Y | X=x)$(b) Use part (a) to find $E(Y) .$(c) Use part (a) to find Var(Y).

(a) 2x, x (b) 40 (c) 100

Intro Stats / AP Statistics

Chapter 4

Joint Probability Distributions and Their Applications

Section 1

Jointly Distributed Random Variables

Probability Topics

The Normal Distribution

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Oregon State University

Idaho State University

Lectures

0:00

01:20

A research biologist has s…

05:49

Finding Area Find the area…

05:35

Plant Growth Researchers h…

05:04

Sketch the region enclosed…

02:29

04:06

In a fish farm, a populati…

05:54

Finding area Find the area…

06:32

If $f(x, y)=\left(10,000…

04:59

Fish Population The popula…

05:13

02:50

$5-10=$ Sketch the region …

09:17

Sketch the region bounded …

02:00

Biologists often use recta…

03:55

Regions between curves Ske…

05:26

Let $X,$ the number of fla…

02:54

The spread of a contaminan…

09:24

Using a Sphere Find the ar…

01:02

Solve each problem.A r…

07:35

04:20

Find the area of the regio…

So given that X is exponential distribution with me. So we have with maintain therefore we are going to get a pdf of F of X being equal to 1/10. We have each the ball -1 for weapon. So access radius so area of secular region. So we have area, right? Yeah, yeah, yeah. Area of circular region to be ego too. Okay guys exploit sooner. We move on to a part a of the question. Sophie So there are 0.5 plants pay square feet. So average number of plants in the given in the region is going to be ego too. So we have London being equal to zero brains. Life multiplied by by bye ex spread. So now we know um why is Watson? So if, why it's possible then you're going to get expectation of why given X is equal to X to be equal to London. And here we know what and the is we know land is 0.5 time is five times X squared. So we are going to get zero planes five. I want to ask hi ex, quit. Right? And for our variance which is via of Y given X is equal to X is also equal to London because we know for amazon they mean Standard and the variances lander. So here too we are going to get 0.5 pi lambda squad. So for people that to be yeah. So how we pass, you're supposed to find expectation of white mhm. And here we know it's going to be equal to have expectation of X into records expectation of Why Giving X. Which is before two x two You got EX over here and we know the answer for this. So we are going to just substitute this in here to know simplifying this. You're going to get zero Greens 5. We have the value of which is the variance of X chris um I mean squared right? So we have 0.5. You know the variance of eggs to be go 200. You know what I mean? Yeah, it will be 100. So you're going to get what's applying this is what we have. I'm running back over here. We have price. Sorry. So we are going to get several points five London. Thanks 200. So this is going to be equal to we have hand. Right? Right. Yeah. For the birth. So for our last part which is the seabed. Yeah. So see we have the variance why? And for this one is going to be equal to have ever of So open a rocket. We have expectation of white giving X which is equal to X small eggs class the expectation of to open another bracket. You have the experience of Why given X which is equal to X. All right. So all this is going to give us and we have the variance of You know the answer for this to be zero point right? Um Hi X squared and here to know the answer already. You know what is to be equal to To they're going to get 0.5 I. X. Squared over here. So simplifying this By the parents. We are going to get zero for him to five. Yeah. Very sweat. Then the variance of X squared Plus zero Greens 5. Okay. You have expectation of X. Grade. All right. So this will be called to 0.25. Mhm. Try sweat when you open the brackets. We have expectation of extra power for grayness. We have expectation of X. Quaid or they squared. So this is going to be equal to no. So expectation of X squared all squared then plus people get the bodies last 100. Bye. So you're being tickets and zero 3 to 5 guys quit. Then in a rocket we have from bacteria. Then I stand suitable for I mean there's 200 square plus 100. Hi. So I didn't order this. You have going to get So either or this. Two other orders in the bracket and multiplying it by 0.25 price. Great. You're going to get 501,100. Okay.

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