00:01
Okay, so we have these random variables yn, which are defined as 1 plus n times xn, where xn are banuli random variables with mean 1 over n.
00:10
They take the values 0 or 1, and the expected value is 1 over n, which is the same as the probability that xn equals 1.
00:19
And we want to show that the random variables yn converge to 1 in probability.
00:24
So what does this mean? this means that for all c greater than 0, so for 1.
00:30
Any positive real number, the probability that yn minus 1 is greater than c converges to 0 as n converges to infinity.
00:43
So firstly, we fix c greater than 0, and then we need to show that this converges to 0 as n tends to infinity.
00:51
So using the epsilon definition of convergence as n tends to infinity, we want to let epsilon be greater than 0.
01:00
And then we want to choose some capital n and we'll come back to this so we're going to choose capital n and we'll come back to this when we see a good condition to to use and then for all n greater than or equal to this capital n let's look at the probability that y n minus 1 is greater than c and we want this to be less than epsilon and this will this by definition means that this converges to zero.
01:33
Okay, so looking at this, this is equal to the probability.
01:38
Now let's just plug in the definition of y n.
01:41
Yn is 1 plus n times xn, and then we have minus 1 greater than c.
01:47
The ones cancel, so this is the probability that n xn is greater than c.
01:53
Now remember xn is a banuli random variable, so it either takes the values 0 or 1, so we don't really need this absolute value here.
02:01
We can get rid of this because this is always going to be non -negative.
02:06
So this is the probability that n x -n is greater than c...