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(6 points Let the random Variable X be GJaussian with mean and variance 0" X ~ Nl/.6?). Consider the random variable X Calculate the probability density functi...

Question

(6 points Let the random Variable X be GJaussian with mean and variance 0" X ~ Nl/.6?). Consider the random variable X Calculate the probability density function of Y. fx(W) first calculating the cumulative distribution function Fy(y) and then taking the derivative of Fx (u) Calculate the probability density function of Y. fxly) by using the general strategy of calculating the Jacobian

(6 points Let the random Variable X be GJaussian with mean and variance 0" X ~ Nl/.6?). Consider the random variable X Calculate the probability density function of Y. fx(W) first calculating the cumulative distribution function Fy(y) and then taking the derivative of Fx (u) Calculate the probability density function of Y. fxly) by using the general strategy of calculating the Jacobian



Answers

Find the expected value and variance for each random variable whose probability density function is given. When computing the variance, use formula (5). $$f(x)=\frac{1}{18} x, 0 \leq x \leq 6$$

Proper 14. We want to identify this probability dynasty function over the center. Then find the mean the variance and the standard division. Without integration we can rewrite F of X to be equal. Two divided Britain. Then simplify. It equals one divided by five. We can rewrite one divided by five to be one divided by five minus here. As we can write F of X As one divided by B minus A. Then it's only from distribution because B equals five and they equals you. Then you have X represents a uniform probability density function. Because it's a uniform distribution then you equals half, multiplied by a plus B equals half, multiplied by zero plus five Equals half, multiplied by five gifts 2.5. And for the variants it equals One divided by 12, multiplied by B -A. To the power of 12. Sorry. To the bar of two Equals one divided by 12. The boy boy B. It's just five. And this a which is zero sq Equals 25 divided by 12. And to get the standard division, we get the square root of the variants it equals, it's sigma Square root of 25, divided by 12 equals why, Divided by square root of 12? or in decimal mm Equals. 1.4.4 city needs to find an answer to our problem.

In this problem, it's required to find the value off New X and you Why? Let's start by New X. Your fax is integration off our boundaries from 0 to 1 for X and Y six exports three to play. Why the X do you want will be equal integration from 0 to 1 three over toe? Why do you, Roy, which is equal to 3/4 and for m y or mule boy? Thesis is mule Milton knew why so Integration from 0 to 1 seeks experts toe Why poor toe g x d y all these years? Thanks. So it will be integration from 0 to 1 two boy pose to do y which is equal to over three. So me, why is two or 3 a.m. u X is 3/4. Thank you.

Today we're going to solve a problem. Number seven here ffx Because to elect Scinto one minus X, the whole square zero does not. The context does not equal to one. Expectation is equal to into the 0 to 1 eggs in do to elect minus 24 X squared plus dual x cubed the X Well, Intel 0 to 1 ex clarity X minus 24 It was 0 to 1 picks cube. The yes do well in the 0 to 1 existed for the it's, which is equal to for execute 0 to 1 minus six Xs 24 0 to 1 plus well accessed. If I buy fight 0 to 1 but difficult to foreign toe. One cube minus four into zero Q minus 16 to 1 just before plus six into zero. Rested for plus you really do. One goes to fight by fight minus two Will in due the road is 25 by five, which comes to be by faith. Then we need to find ways and soft X variance off X is equal to the road to one eight square daughter to elect minus 24 X square plus well X cubed DX minus Dubai Side, the whole square. So which comes to be like well and 0 to 1 x cubed de Explain US 24 0 to 1. Access to for the X plus well, Intel 0 to 1. Access to fight the X minus four by 25 which is a call to three acceptable by existed for from 0 to 1, minus 24 access defied by five from 0 to 1 plus to exist to six 0 to 1, minus four by 25. So which comes to be like three minus zero, minus 24 by fight zero plus two minus zero minus four by 25 which is a kowtow won by 25 Expectation all fixes one way flight Parents off axis One day 25. Thank you.

Yeah it's probably been given the following joint distribution. We would like to find the cool variance. So that's what now the co variance is equal to the expected value of X. Y minus the expected value. That's times the expected value of what? Finding this in pieces here. The expected value. That's why is the double integral. That's why times are joint distribution And notice that X&Y both go from 0 to 1. And we're going to evaluate these both from 0 to 1. Now what? You neither evaluate these my hand or type this in to a graphing utility in order to grab them Or excuse me in order to evaluate this and I'm gonna use a graphing utility here and doing so gives us the expected value. That's why is one third now finding the expected value of X. And the expected value of what? Uh huh. The expected value of X will be the double in a row of X times X plus Y. With why knowing from 0 to 1 an X. Coin From 0 to 1. Similarly the expected value of Y will be this interval. What's up with Y. Times X plus Y. All right. Mhm. Now in order to integrate this notice that these who are the exact same thing. The only difference is that actually replaced with why? But they're going to give us the exact same value which is they are symmetrical And so we just need evaluate one of them and then they would both give us the same value evaluate this top because 7 12. So that bottom is also 7 12. So come back up here to our formula. Mhm. We have the expected value. That's why minus the expected value. That's times the expected by of why. And this is 1/3 -7/12. Uh huh. Times 7/12. Mhm. This is the expected value. That's why and this is the expected value events and they expect the value of why. Pleasure. And so evaluating this gives us our co variance, Which is negative 1/1 44 In circumstances that you have won over 144.


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