Let X, Y be continuous random variables with joint...

Name: Let X, Y be continuous random variables with joint probability density function (pdf) f (x,y) = Cx³ (x - y), 0 < y < x < 1. (a) Find the value of C. (b) Find the marginal pdfs of X and Y. (c) Find the conditional pdf of Y given X = x. (d) Compute P (0 < Y < 0.1 | X = 0.2) (e) Are X and Y independent?
Uploaded: 2023-08-04T14:48:25-08:00
Duration: 5 min 19 s
Channel: Sri K
Description: Let X, Y be continuous random variables with joint probability density function (pdf) f (x,y) = Cx³ (x - y), 0 < y < x < 1. (a) Find the value of C. (b) Find the marginal pdfs of X and Y. (c) Find the conditional pdf of Y given X = x. (d) Compute P (0 < Y < 0.1 | X = 0.2) (e) Are X and Y independent?

Let X, Y be continuous random variables with joint probability density function (pdf) f (x,y) = Cx³ (x - y), 0 < y < x < 1. (a) Find the value of C. (b) Find the marginal pdfs of X and Y. (c) Find the conditional pdf of Y given X = x. (d) Compute P (0 < Y < 0.1 | X = 0.2) (e) Are X and Y independent?

Transcript

00:01 Hello everyone, let us look into the equation.

00:03 Here we can have x, y it is a bivariate continuous random variable.

00:13 So, f of x, y is c x power 3 less than 1 and 0 otherwise this is the function given.

00:21 Now, in the first one we have to find c value.

00:29 So, take 0 less than x less than y less than 1 is the same as 0 less than x less than 1 and 0 less than y less than 1.

00:42 Now, we have to calculate the double integral f of x, y dx dy which is equal to c times double integral x cube x minus y dx dy and the values can be taken as from 0 to 1.

00:57 Now, we have to take this as equal to 1.

01:00 Now, we have to integrate it.

01:02 So, which is c times integral over 0 to 1 first we will integrate with respect to y.

01:09 So, we may get x power 4 y minus x cube y square divided by 2 from 0 to 1 dx equal to 1.

01:17 So, which is c times integral 0 to 1 x power 4 minus x cube dx which is equal to 1.

01:25 So, which implies c times x power 5 by 5 minus x power 4 by 4 from 0 to 1 equal to 1.

01:34 So, which is c times 1 by 5 minus 1 by 4 which is equal to here we make it as x cube by 2.

01:44 So, it is x power 4 by 8.

01:47 So, 8 minus 5 is 3.

01:49 So, c times 3 by 40 which is equal to 1.

01:54 Hence, we can have c equal to 40 by 3.

01:57 Then next one we have to find the marginal pdf of x for 0 less than x less than 1 and 0 less than y less than x...

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