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Point) Suppose that the moment generating function of a random variable X is Mx(t) = exp(4e' 4) and that of a random variable Y is My(t) = (; & + 3) If X a...

Question

Point) Suppose that the moment generating function of a random variable X is Mx(t) = exp(4e' 4) and that of a random variable Y is My(t) = (; & + 3) If X and Y are independent; find each of the following:(a) P{X + Y =3} =(b) P{XY = 0}(c) E[XY] = 44.8(d) E[(X + Y)2] = 237.28

point) Suppose that the moment generating function of a random variable X is Mx(t) = exp(4e' 4) and that of a random variable Y is My(t) = (; & + 3) If X and Y are independent; find each of the following: (a) P{X + Y =3} = (b) P{XY = 0} (c) E[XY] = 44.8 (d) E[(X + Y)2] = 237.28



Answers

If $X$ and $Y$ are the independent random variables for $B\left(5, \frac{1}{2}\right)$ and $B\left(7, \frac{1}{2}\right)$, then $P(X+Y \geq 1)=$ (A) $\frac{4095}{4096}$ (B) $\frac{309}{4096}$ (C) $\frac{4032}{4096}$ (D) none of these

In this question, in part a we have to find out the MGF means the moment generating function for the Bernoulli random variable by one. So M Y one T. That equals to expectation of E t v i one. This is the definition of moment generating function. Now we apply submission as this is discrete distribution. So submission X is equal to 0- one. E t V I be right. The system mass function of the Bernoulli distribution, which is equal to p Y is equal to B Y one minus B one minus Y. Where Y is zero or 1. Now we put the function E t y is equal to p vie 1 -7 1 -6. Now we put the value zero and 1 in this function. Then it's simplified to ET0, one minus B Plus ET one B surf woman generating function of Ben ali distribution becomes one minus P plus p E T. Now, in part B we have to find out the moment generating function for w where W is equal to By one plus Y. two plus so on. Why? And so M W T. That equals to expectation of E tw Submission 1 to end. I am why I. D. It becomes as y as I is equal to one to so on. And So our function becomes one -P. Class B E t well power. And since the MGF or w is in the form offer binomial and GFP with end trials and success probability P. So this is the distribution for W. Thank you

In this question, in part a we have to find out the MGF means the moment generating function for the Bernoulli random variable by one. So M Y one T. That equals to expectation of E t v i one. This is the definition of moment generating function. Now we apply submission as this is discrete distribution. So submission X is equal to 0- one. E t V I be right. The system mass function of the Bernoulli distribution, which is equal to p Y is equal to B Y one minus B one minus Y. Where Y is zero or 1. Now we put the function E t y is equal to p vie 1 -7 1 -6. Now we put the value zero and 1 in this function. Then it's simplified to ET0, one minus B Plus ET one B surf woman generating function of Ben ali distribution becomes one minus P plus p E T. Now, in part B we have to find out the moment generating function for w where W is equal to By one plus Y. two plus so on. Why? And so M W T. That equals to expectation of E tw Submission 1 to end. I am why I. D. It becomes as y as I is equal to one to so on. And So our function becomes one -P. Class B E t well power. And since the MGF or w is in the form offer binomial and GFP with end trials and success probability P. So this is the distribution for W. Thank you

Therefore we want to find the cumulative distribution function F of X for the random variable X. And the sketches to griff with the probability function exotics For the following probability functions, the first one the xbox X equals one Where X equals zero. This is the only value of X that has probability. Then assembly F of X equals the submission B of X, which equals one When x equals zero. By graphing this. We just substitute bye F of X or B of X to be one At x equals zero and it's else equals zero elsewhere. But F of X, it's not zero elsewhere. You can through it using another colour. Because from the definition of F of X equals zero Before x equals zero and equals one after X equals E. Then the red line represents the function ex a cumulative distribution function. And the blue line represents the probability function. Let's get to barbie, B X of X equals one third, Where X is -1, 0 and one only then to find F of X. We can't find it in four intervals. When X It's smaller than -1, it equals you and between minus one and zero X to be between and so on and zero. It's defined to be one thing only 1/3. When we get X greater than zero and smaller than one, it's two thirds And finally it equals one for ex Later than equals one. Then we add Equal sign with zero. We had equal sign with minus one. And here in party to probably define F of X. We should define it using this way, two intervals When X is smaller than zero and when X is greater than equal Z. Here it's fine and zero and here is fine as one. That's the graph barbie. Here is X. Here is a full backs, Has a maximum value of one. A value of -101 Before minus one, F of X equals E. Then takes a step To be at 1/3 until we reach is you When you reach zero, takes a step To get 2/3 To reach one. When we reach one Takes a step to get that one to infinity. This value is 1/3. This value is stupid. This is for F of X. Let's drop P of X just only one third At -1-0-1. It's 1/3 at x equals -1. It's zero, It's 1/3 that equals zero. It's 1 3rd tax equals one And it's the fine zero elsewhere from minus and 22 and 20 except at -10 and one. Let's get to bar. See B of X is defined as x divided by 15, Where x equals 1, 2, 3, 4 and five. Again well defined F of X insects intervals When X is more than one. When X equals warm To one of them too. From two 2, 3 from three, 24 from four, 2, 5. And finally X greater than equals fine. Here it's zero. Here we take X divided by 15. Where x equals one, then it's one divided by 15. In this interval we add The probability at x equals two. We add two divided by 15 and three divided by 15. Here we add be of three. We have three divided by 15. It's six divided by 15. Here we add probability of X equals four. We add 40 by by 15, standby by 15. Finally we add B of X equals five. It's 15 divided by 15. This is F of X. Mr graphic, we have 12345 it's fine. Zero. and then here take a step 2, 1 divided by 15 if this is warm And then one divided by 15 is about here, then it's the street, something like that, then it's six, something like that. Sorry then it's then something like that. And finally one These values one divided by 15, three divided by 15, six divided by 15. This is 10 divided by 15. And finally here at 15 divided by 15. Which is well let's add me of X. B of X is defined as X file by 15. Just the points 1, 2, 3, 4 and five And find a zero elsewhere. And it's zero except at one 2345 at one. It's one valuable 15 to here three years for here five years.

Therefore we want to find the cumulative distribution function F of X for the random variable X. And the sketches to griff with the probability function exotics For the following probability functions, the first one the xbox X equals one Where X equals zero. This is the only value of X that has probability. Then assembly F of X equals the submission B of X, which equals one When x equals zero. By graphing this. We just substitute bye F of X or B of X to be one At x equals zero and it's else equals zero elsewhere. But F of X, it's not zero elsewhere. You can through it using another colour. Because from the definition of F of X equals zero Before x equals zero and equals one after X equals E. Then the red line represents the function ex a cumulative distribution function. And the blue line represents the probability function. Let's get to barbie, B X of X equals one third, Where X is -1, 0 and one only then to find F of X. We can't find it in four intervals. When X It's smaller than -1, it equals you and between minus one and zero X to be between and so on and zero. It's defined to be one thing only 1/3. When we get X greater than zero and smaller than one, it's two thirds And finally it equals one for ex Later than equals one. Then we add Equal sign with zero. We had equal sign with minus one. And here in party to probably define F of X. We should define it using this way, two intervals When X is smaller than zero and when X is greater than equal Z. Here it's fine and zero and here is fine as one. That's the graph barbie. Here is X. Here is a full backs, Has a maximum value of one. A value of -101 Before minus one, F of X equals E. Then takes a step To be at 1/3 until we reach is you When you reach zero, takes a step To get 2/3 To reach one. When we reach one Takes a step to get that one to infinity. This value is 1/3. This value is stupid. This is for F of X. Let's drop P of X just only one third At -1-0-1. It's 1/3 at x equals -1. It's zero, It's 1/3 that equals zero. It's 1 3rd tax equals one And it's the fine zero elsewhere from minus and 22 and 20 except at -10 and one. Let's get to bar. See B of X is defined as x divided by 15, Where x equals 1, 2, 3, 4 and five. Again well defined F of X insects intervals When X is more than one. When X equals warm To one of them too. From two 2, 3 from three, 24 from four, 2, 5. And finally X greater than equals fine. Here it's zero. Here we take X divided by 15. Where x equals one, then it's one divided by 15. In this interval we add The probability at x equals two. We add two divided by 15 and three divided by 15. Here we add be of three. We have three divided by 15. It's six divided by 15. Here we add probability of X equals four. We add 40 by by 15, standby by 15. Finally we add B of X equals five. It's 15 divided by 15. This is F of X. Mr graphic, we have 12345 it's fine. Zero. and then here take a step 2, 1 divided by 15 if this is warm And then one divided by 15 is about here, then it's the street, something like that, then it's six, something like that. Sorry then it's then something like that. And finally one These values one divided by 15, three divided by 15, six divided by 15. This is 10 divided by 15. And finally here at 15 divided by 15. Which is well let's add me of X. B of X is defined as X file by 15. Just the points 1, 2, 3, 4 and five And find a zero elsewhere. And it's zero except at one 2345 at one. It's one valuable 15 to here three years for here five years.


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