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Point) The random variable X has mass functlonX=iP(x) =x =2X =3Provide your answersdecimal places:Find the moment generating function of X and evaluate it at the po...

Question

Point) The random variable X has mass functlonX=iP(x) =x =2X =3Provide your answersdecimal places:Find the moment generating function of X and evaluate it at the point 1=2.Find the second derivative of the moment generating function and evaluate it at the point (=2_Use the m.g.f. to find the second moment of X,

point) The random variable X has mass functlon X=i P(x) = x =2 X =3 Provide your answers decimal places: Find the moment generating function of X and evaluate it at the point 1=2. Find the second derivative of the moment generating function and evaluate it at the point (=2_ Use the m.g.f. to find the second moment of X,



Answers

Refer to Exercise $3.145 .$ Use the uniqueness of moment-generating functions to give the distribution of a random variable with moment-generating function $m(t)=\left(.6 e^{t}+.4\right)^{3}$.

Now here on this problem, we are told that why is a normally distributed random variable? And we want to do some things with X equals negative three Y plus four. First we want to find the moment generating function. Now from exercise 4.137, we know that the moment generating function of a Y plus B is equal to either the TB times the moment generating function of Why evaluated at 18. Now, what this means? Course here is at the moment generating function of X. There's a moment generating function of -3. Y Plus four. So as you can see B is four. And so this means this is equal, it's either the fourty times A is -3. So this is the moment generating function. And why Evaluated at -3 T. Yeah, This is either the 40 And then we plug in -3 T. The moment generating function that they'd give us. And so this is E To the -3 mut. Yeah. Plus one half negative three T squared sigma squared. And so this is equal to E. To the -3 μ plus four T plus one half 90 squared cinema square mm. And so this is our moment generating function correct Now. We also want to know what the distribution of exes now notice that this is the MGF of a standard normal. Not a standard normal. Just a normal. Excuse me of a normal. Yeah. With mean of negative three mu plus four. That's always what's on that T there? That's always your means. That's 93 mu lost four in the sigma squared of nine T squared sigma squared. Again, that's always what's up top here. Not t squared. Usually just nine sigma squared. Okay, your sigma is always your variance is always what's on that, the one half t squared there. And that means that x is normal with a mean of negative three mu past four and the variance of nine sigma squared. And that's just comes from its moment generating function.

So in this question, be a given that of. So in this question it is said that the empty moment about some point. We so some point B is given the expectation of expanded this to him. So this country is called the moment about B. And the question asks, what is the 1st 2nd? No question gives us the 1st, 2nd and 3rd moment about seven And those are 3, 11 and 15. It asks us to find me and the 1st 2nd and 3rd moment about me. So to do this question will write the 1st, 2nd, 3rd moment in terms of About new, in terms of the 1st, 2nd and 3rd moment about seven and then calculate Uh these 300 ish. So we know that expectation of x minus seven people 23 Since the first moment about 73. This gives us expectation of X -7, seven is a constant equal to three. Therefore expectation of X comes out to be 10. So this is the first answer. So you use them. Next. The first moment about then will be expectation of X -10. This will be for the expectation of X- Tan is equal to 10, you understand zero? So first moment about 10 would be zero. So next we have expectation of X -7 Squared. This can be written as expectation of x minus 10 plus three. The old square then we separate out by expanding so x minus 10 squared plus nine plus two times 3 to 6 times x minus 10. So expectation of X -10 square, That's nine is a monster on the extent expectation of X -10. But expectation of X -10 is the first moment about about them Which came on to be zero. Yeah so this time will drop out. And expectation of X 27 year old squares 11 because the second moment about seven So 11 is equal to expectation of X -10. The whole square last nine This implies expectation of X -10 sq will be equal to. So the second expect moment of second moment about you will be mm So for the 3rd part Letter Write X -7 Cube in this from so it's minus 10 plus three. So this will be X minus seven the whole cube. So take this has to be the first time And take 3 to be the second. So this will become X -10. The whole cube. Let's treat you plus three A. B. So three x minus 10 legal victory for T. V. Times A plus B. So X managed 10 plus three So X -7. So basically A plus B. The whole cube is equal to eat you. Let's be cute. Just three Kb A blessed peace. So this is what we're using over here. So Ex Minister Ncube plus nine plus nine times 339 times x -10. And it just be will be so x minus seven. I'm writing again is x minus 10 plus three. So in this expression take x Pakistan to be the first time And 3 to the 2nd time. So you got nine x minus 10 times x minus tan. So nine x minus two whole square plus 9327. It's Madison. So taking expectation on all the sides, you'll get expectation of x minus seven Q. This one is equal to expectation X minister thank you. Last nine plus nine times expectation of x minister in the whole square. The expectation of X Pakistan will be zero which we have already found and party so replacing zero And X -7 Q. This is nothing but the third moment about seven. This is given to be 15. So we know this, this is what we have to find And X -10 square expectation like Afghanistan Square is also known to be 11. So substituting this, we get expectation of x minus T. 10. Q is equal to minus 30. So the answer to part three will be minus 20.

You know this problem? We were told that we have a random variable with the density function of F. Of Y is equal either the why or why is lessons your first. We want to find the expected value of E. To the three Y over to and the expected value of each of the three. While over two is equal to the integral from negative infinity to zero. Because that's when are Y is defined of each of the three Y over to times either the Y. Dy now this is equal to the integral from negative infinity to zero of E. To the five house Y. The wine which is equal to end to the five halves. Why times 2/5 from wyffels, negativity is there And when I plug in zero in for Y. This will give us two fists. E to the zero. When we take the limit of what goes to negative infinity, that would give us zero. Mm. So we have 2 50 to 0 minus zero which is april 2, 2/5. And so far it a it's too fast. No one be. We want the moment generating function. The moment generating function, yes. Is april to the expected value of E. To the T. Y. And so this is equal to the integral from negative infinity to zero of E. To the T. Y. Times either. The white. Yeah. Which is the integral from negative infinity zero of E to the T plus one. Y. Do you want this is equal to one over T plus one, E. To the T plus one, Y. From wyffels, negative infinity zero Paula Deen and zero. We have one over T plus one, you know, zero. And then taking the limit as we go to infinity of this expression when you have zero. So this tells us that the moment generating function is one over T plus one, which is the same as teapots, one to the negative one. Now and see, we want to find the variance and find the variance. We need to first find the expected value and the expected value is equal to the derivative of the moment generating function evaluated at zero now are derivative of the moment generating function is negative T plus one to the negative second. That means that in private zero is equal to negative one. That's what I expected value is negative one. And we also need the second derivative. The second derivative is to times two plus one the negative and so the second derivative at zero is equal to two. And that's the expected value of Y squared. The variance of Y is equal to the expected value of Weisberg minus the expected value of why square. That's why we need to both of those moments directed by of Weisberger's two expected value wise, negative one and then we square, and this is two minus one, which is one of the variants of why this one.

Now here on this problem, we are told that we have a moment generating function. Name of T Is equal to 0.6 either the T Plus 0.4 All to the 3rd power. And we would like to use this in order to find the distribution of this now from problem 3.145. Okay. We found that for binomial random variable, our moment generating function is equal to P. You know the tea, ask you All to the 3rd power. Now, it's maxie's up here. P is what's on the eve of the T notice. We've got a 0.6 on the eve of the T. Until it tells us that P 0.6, Q is 0.4, but Q is just one minus P anyway. And so if you know P you know what he was And then our exponent and that should be a three s. p. n. Our exponent is three. And so it tells us that N is equal to three And so we know P is 0.6 and we know that in is three. And for some people that's enough for the distribution. But if we go ahead and write out the probability of why for binomial this is in choose why. Okay, times P to the Y, times yeah, two to the n minus Y. And so here, since in his three is three, choose why Time 0.6 to the Y. And then Q is 0.4. So this would be a time 0.4 TN -1. So here's our distribution.


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