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Question 1. Consider the experiment of rolling pair of fair dice. If 11 represents the value Ohl the first die and 12 represents the value OH the second die. let Y ...

Question

Question 1. Consider the experiment of rolling pair of fair dice. If 11 represents the value Ohl the first die and 12 represents the value OH the second die. let Y = X1 + Xz and Z = Ii - Xzl: Find the marginal probability mass functions of Y and Z. (10 pts Find the conditional probability that Y _ 7 given that Z = 3 5 pts) Find the conditional expectation of Y given that Z = 3. (5 pts)

Question 1. Consider the experiment of rolling pair of fair dice. If 11 represents the value Ohl the first die and 12 represents the value OH the second die. let Y = X1 + Xz and Z = Ii - Xzl: Find the marginal probability mass functions of Y and Z. (10 pts Find the conditional probability that Y _ 7 given that Z = 3 5 pts) Find the conditional expectation of Y given that Z = 3. (5 pts)



Answers

Find the expected value for each random variable.
$$\begin{array}{|c|cccc|} \hline {z} & {9} & {12} & {15} & {18} & {21} \\ \hline {P(z)} & {0.14} & {0.22} & {0.38} & {0.19} & {0.07} \\ \hline \end{array}$$

Find the probability that if you toss a die twice, this sum is less than 11. So toe, get the sample space fist. We can just use the at a tangle method. And then we have one, 23456 We have one too. 3456 This is the first toast. This is the second tossed, so we can add one plus 12 34567 And we can have two plus 134567 Hates. Then you can always follow the pattern. This is for five, 67 89 Now I can see this is 1234567 Then this is 56789 10 They see 6789 to love him. This is 78 9, 10, 11 and 12. So what we want those are the sun, uh, that you get now, what is the probability that the sum is less than 11? Now, when you look at the Sam's Year, but greater, you can use the sums that are greater than 11 and then do the compliment So those at three therefore we can say use the complement one minus or 36 over 36 which is one minus three over the sample space, which is 36 which is 36. Ministry is steady. Dream Over 36 which can be reduced to its lowest team, is 11 over off by dividing by three, So the probability that the sum is less than 11 is 11 off a 12.

All right, So this problem, we're gonna deal with some probability. So let's start with a Sometimes we have to dice and it's asking us for the probability Rolling five. So write out all the different positive with these year of actually getting a five so we can make a table dice one that's too. So we could roll 14 to three Korean, too, for and a lot. So that means we have four possibilities. Roll your five and there are 36 total possibilities. Um, I've come. There's 36 total combinations that we could get one willing to dies. So we have to do is to for over 36 which is the same thing as one over nine. So that's it. For part, a Part B is similar. It's asking the probability of rolling in 11. So once again, we can do our table here. It's gonna be even smaller this time because the only possibilities 56 65 So now we only have to. So we're going to your two out of 36 which simplifies down to one out of 18

So here we're throwing two dice and two dice allows for 36 possible micro states and this is really easy to calculate because if you have a six sided die, it's simply going to be sick side six sides to, uh per die. And then we have to the second power because we have to dice. So this legal 36 that's how we find in the micro States and then part A. We want a sum of seven. So that would be one and six. It would be two and 553 and four, four and three, five and two and six and one. So that would be 12345123456 So sick. So I guess the problem rather the probability would be equal to 60 136 where you can say one over sex for B, we have we rather we want some of 11. And here there are actually on Lee to so five and six and we have six and five Ah, giving us a probability equal toe to over 36 or weaken. Say, ah, one over 18 and then lastly for C, we want a sum of four and here. There's only one. Well, rather we have. Sorry. One and three, two and two. And then three and one. And so here. The probability would be three over 36. Probably equals three over 36. And in this legal, of course, one over 12. So these would be our three probabilities. That is the end of the solution. Thank you for watching.

So we're justifying the average. And just by looking at this visually we see that it's equally symmetrically distributed around seven, meaning that seven is our average, so average Is equal to seven. Ah But now for part B we have to obtain the standard deviation which is going to require a little bit more work as a reminder this is the formula. And we're gonna go ahead and figure out the inner part right here of the samba X. Squared P. X. Equals X. So just the probability that has equals that. Except we're gonna be looking at the wise. So these should really be wise instead of excess. Mhm. Like this. All right. So first one is gonna be two So it's gonna be two squared Times 1/36 plus three squared Times on over 18 plus four squared times 1/12. Okay. And this keeps going on Until he reached 12. Whereas 12 squared times 1/36. She's gonna be a pretty small number. And then we subtracted from the average, we're gonna get some value here, we're gonna call that a so the standard aviation is going to be equal to a minus seven squared 49 And this is gonna be equal to about 2.4. And then they want us to drop our ability instagram. Okay, so it's gonna be pretty easy just gonna draw our access, this is the sum, but why? And then the uh wait this is the frequency and this is the light and we're going to have plenty of bars here, it's gonna go like this up here, this is just going to keep increasing. And how many do we have? So we have 10, 12345 So that's gonna be the highest one and it's gonna look perfectly symmetrical that the average is gonna be right here, and and the standard deviation is gonna be 2.4 out, So it's gonna be a little bit like this, that's one standard deviation away, that's two standard deviations away


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