Okay, so we have this fx of X. Which is a poison distribution. We're land as equal to 100 and were also given the F of Y given X distribution which is a binomial distribution With a probability of .6. And so what we're gonna do with those two is we're going to multiply them together to find the joint distribution probability of X and Y. And then we're going to add all values of X. Of our of our product which is a joint distribution. And that will give us the actual distribution of why. And so what we're gonna do is follow this equation I have written here which is the summation of the joint distribution of all values of X. And we're going to use that to find the distribution of Y. So this is actually equal to if we plug in our terms, we have the summation from X is equal to Y to infinity of E. to the negative. 100, 100 Times 100 to the x divided by X. Factorial. And then if we plug in our conditional probability F given F of Y given X, we have X factorial divided by Y factorial times x minus Y factorial multiplied by .6, Y and one minus 10.6 is 0.4. So this is multiplied by that and this is multiplied by .4 X minus Y. And now what we can do and so we have this all written out, not for the multiplication, there is we can take out all the terms that aren't affected by our summation. So all the terms that don't have an X in them. So that would be U. to the negative 100 and Y factorial And .6 Y. And then what we're left with is the summation of X is equal to Y to infinity of 100 x 0.4. The x minus Y divided by x minus Y. Factorial. And I also have cancelled out these X factorial since we have one on the top and one on the bottom. And so this is what we're left with. And now what I'm gonna do, So I'm actually gonna make this .6, Y. I'm gonna make it 60. Why? Over 100 Y. And then I'm gonna put the 100 wide back into our summation. So what it's gonna look like is Each the negative 100 times 60 y divided by Y. Factorial times the summation of 100 x Divided by 100 Y. And that's our That's the 100 why I took from this 60 divided by 100 Y. And put it back into our summation. No safe from X is equal to Y to infinity. And then I'm going to multiply this by .4, X -Y and divide by x minus Y. Factorial. And now that I have this 100 ext about 100 Y. I can put this into one exponent by minus ng, the Y from the X. And so we have and then after I have that, why that 100 x minus y, I can multiply it by 1000.4 x minus Y. So we have 100. Each of the negative, 160 to the Y multiplied by the summation of 100 X minus y. Times 1000.4 x minus Y is 40 X minus Y divided by x minus way factorial. And now this is where we can use the MacLaurin series for each of the X. So if you're say to let x minus Y equal, say, and so you say n is equal to x minus y. Then we can rewrite rewrite this summation so that it's the summation of N is equal to zero to infinity of 42. The N divided by and factorial. And now if we look at this, if we look at the um MacLaurin series for each of the X. MacLaurin series for each of the X is equal to and is equal to 02 Infinity summation from N is equal to zero to infinity of X to the end over N. Factorial. And now since we have our summation in this form we can instead of using X we can just plug in this 40. So we have the summation from N is equal to zero infinity of 40 to the N divided by N. Factorial. That's actually equal to eat the 40th using this MacLaurin series. And then we just plug this back in and find our distribution. So Once we make the substitution of each of the 40th for our summation we have Either the negative 100 multiplied by 60 Y. Divided by Y. Factorial. And then we have this multiplied by E. to the 40th. So Times E. To the 40th. And now what we can do is just add the exponents of E. So negative 140 is negative 60. So we have F. Of Y. Why is equal to Eat the negative 60th Time 60 to the Y. Divided by Y. Factorial. And now if you look at this, a poison distribution with X. And lambda is equal to E. To the negative lambda landed to the X. Over X. Factorial. We can see that this poison distribution formula actually does. It looks exactly like our F. Y. Of Y probability distribution or probability mass function. So we do indeed have a poison distribution for our Why variable. And now all we have to do is find the expected value in the variance. And for any poison distribution, the expected value, it's just equal to lambda. So in this case are lambda is 60 so this is equal to 60 and our variants Is also equal to λ for poison distributions, so this is also equal to 60.