00:01
Here we're taking a random walk on the plane where at each step you randomly pick one of four moves.
00:10
Either go north, east, south, or west.
00:14
Now we have a random variable, which we call d.
00:20
And d is the distance from w or the distance of the walker from the origin after two steps.
00:28
So the question is, what are the possible places this walk could end? so we could take two steps west.
00:39
So first we go one, two.
00:41
We could also take two steps north.
00:44
Oh, sorry, two steps east was the first one, north.
00:47
We could take two steps east, sorry, two steps west, gosh, and two steps south.
00:55
So there's one way to reach each of these points.
01:01
Okay.
01:03
What if we went east and then north, then we could reach this point.
01:08
And in fact, there's two ways to reach this point.
01:10
We could also go north and then east.
01:14
And it turns out that we have a similar case for every point, sort of around the south side.
01:23
So we could either go east and then south or south and then east.
01:27
And then finally, we could end up actually exactly where we started.
01:31
And there's exactly four ways to do this in two steps.
01:34
We just go east and west, north and south, etc.
01:40
And so if we add up all these numbers, we see that that describes all 16 possible walks.
01:47
And so now we need to talk about this random variable d.
01:52
So the sample space of this random variable d is going to be zero.
02:07
Okay, and then we need to say, what is this distance? well, if we walk one in one direction and one another, then the distance is going to be a square root of 2.
02:18
And then, oh, that's the wrong closing bracket.
02:21
Then we could also land distance 2.
02:24
So you'll see that the possible distances are just 0, root 2, and 2 away from the origin.
02:31
Okay, then the next question is what is the density function? so we can write this as f sub d of x is going to equal.
02:44
0 with probability 4 over 16, sorry, excuse me, this equals 1 over 4 if x equals 0.
02:58
I got it backwards.
02:59
So if x equals 0, the probability of that is 1 over 4.
03:03
What is the probability of the distance being root 2? well 2 plus 2 plus 2 equals 8 divided by 16 is 1 1 1� if x equals root 2.
03:15
There are 4 ways to have distance 2.
03:18
So 4 over 16 is 1 4th, and 0 otherwise.
03:27
I guess we'd call this probability mass function, i would think.
03:31
But anyway, the probability density function is non -zero for exactly three values.
03:36
Those values are the members of the sample space.
03:40
And then the function outputs the probability that we get that value.
03:45
Okay, so the next question is, what is the expected value of d? so the way we do that is we're just going to take 0 times 1 fourth, which we don't have to evaluate, and we're going to get root 2 divided by 2 plus 2 times 1 4th.
04:07
So what do we end up with? we end up with root 2 plus 1 divided by 2.
04:14
That's the expectation of d.
04:17
And then we need to also find the variance.
04:20
So the variance of d is going to equal the expectation of d squared minus the expectation of d squared...