A north american manufacturer produces items with a...

Name: a north american manufacturer produces items with a rectangular base let random variables denote the respective length xy x and y are independent with the following parameter values ex2varx0 94698
Uploaded: 2021-10-10T17:12:33-08:00
Duration: 7 min
Channel: Christopher Dzorkpata
Description: a north american manufacturer produces items with a rectangular base let random variables denote the respective length xy x and y are independent with the following parameter values ex2varx0 94698

A North American manufacturer produces items with a rectangular base. Let random variables denote the respective length (x)Y x and Y are independent with the following parameter values. E(x)=2,Var(x)=0.12,E(Y)=9,Var(Y)=0.37 The perimeter is then defined as 2x+2Y. (Round your answers to 4 decimal places, if needed.) (a) What is the mean of the perimeter of the rectangle? (b) What is the standard deviation of the perimeter of the rectangle?

Transcript

00:02 Hello, so this is the information we've been given the question.

00:05 So i'm going to say that let d be equal to the perimeter and thus is giving us 2x plus what 2 .y.

00:18 This cross -checked that.

00:21 Okay, so that's what we got.

00:23 So for the first part of the question, where we want to determine the mean of the perimeter.

00:33 So that's going to be the expectation of d.

00:44 I'm going to be 2x plus 2y.

00:55 Now let's remember that the expectation of a x.

01:05 The same as that's going to be the same as a times the expectation of x.

01:14 So you're going to be using this property here.

01:18 So this i can rewrite as the expectation of x plus the expectation of let's say it's 2y and using this property right here it's going to be the same as 2 times the expectation of x plus 2 times the expectation of y is going to be 2 times the expectation of 4 plus 2 times the expectation of y is 8 so this is going to be 8 plus 8 plus 16 not just going to be 24 so the mean is 24.

02:19 So look at the second part.

02:22 We're doing for the standard deviation.

02:26 So the standard deviation, i'm going to be using the variance here.

02:30 So the variance of d is going to be the variance of 2x plus 2y.

02:42 And this is going to expand to the variance of 2, oops, the variance of 2x plus the variance of 2y.

02:55 2y plus 2 times the covariance 2 and 2 y so let's see so here we need to know that x and y are independent x and y are independent so the implication of that is that the covariance of 2x 2 y because they are independent that's going to be called zero so what you're going to have now now, the variance of d is going to be equal to the variance of 2x, 2x plus the variance of 2y.

04:03 We're going to use that same property.

04:07 That's going to be, let's see, just a second.

04:20 I'm trying to figure out something here real quick.

04:29 Alright.

04:31 Okay, so that's going to be 4 times the variance of x.

04:42 Because remember that the relationship between variance and standard deviation is the variance squared...