00:01
Now here in this question, we basically have two discrete random variables, and the joint, probably math function, is given in this table, right, as given here.
00:11
And you ask you to calculate the marginal probability of mass functions, pxx and py.
00:16
Well, that is, so let's look at px first, px, and of course, that is simple, the sum over all the joint, the joint distribution over y, right? over y, that's it.
00:33
So let's do it.
00:36
We will find that we need to look at three possibilities, actually.
00:40
First, look at x equals 1, right? x equals 1.
00:44
You sum over y, x equation, some of why you put them with these three numbers together.
00:48
You get 0 .5, right? and similarly, if you sum for x equals 2, you sum this three, three numbers together, you get 1 .2.
00:58
So that's for x equals 2.
01:00
And finally for x equal 3, you put these three numbers together and you get 3 and that's x3, right? so that's a margin distribution pxx.
01:10
And similarly, you can find p y y, right? so you just knew some over x actually over some of those joint distribution over x possible values.
01:24
And then y, again, you have to look at three possible plus plus plus plus y, when you look at y -equish one, you sum over these three numbers.
01:29
You get 0 .2.
01:32
Now, y -equals 2, you sum over these three numbers.
01:35
You get it from 3.
01:38
And finally, when you're y -equal 3, you can sum over 3, and you get 0 .5, right? so that's marginal distribution functions.
01:48
How about other random variables in the pan? the answer is you know, because you look at the, if you look at the distribution, and obviously the margin distribution depends on, you know, the x and y, the x value, right? for example, you look at the px, it does depend on the value of y, right? the value of, sorry, if you look at the, if you look at this, john, if you look at this table, and you look at, for example, for x equals 1 and for x equals two, it's very clear that the distribution is different for y, right? so that means the distribution of y actually depends on the value of x.
02:31
And similarly, you look at these two rows, the distribution of x depends on the value of y.
02:37
So they're not independent.
02:40
So the answer is that no, they're not independent.
02:47
So you also need to calculate this quantity, the probability for this event, that x is smaller than one, but x is smaller than one.
02:58
An x is more than two, but you just need to calculate x more than one because for this happening, obviously, you just need x less than no, you need x less than two, right? so basically what this event is that what's probability, okay, so this is a conditional probability, right? so what this means is that what is the probability for x less than one and if x already less than two, right? if you look at that, if you look at x less than two, less than two, so you have, if i look at x less than two, that means basically look at this two, right? right.
03:39
Look at these two columns, right? so there's two columns.
03:43
You look at x less than two, what's probably for x less than two to happen? why you have to sum over all of these numbers, basically.
03:50
That's what's happening, right? and that's actually point one, point one.
03:57
And then this is, so you're all together, you get a point seven, which is, and then you have to, you have to use that point saving as the denominator...