00:01
Here we have 10 samples.
00:04
Let's see.
00:05
X1 to x10.
00:10
There are 10 samples from an exponential distribution with mean value 2.
00:19
I mean, there are iid random variables with exponential distribution with parameter 1 half because let's say y is a random variable with distributed exponentially distributed with parameter lambda then we say the density function for y is equal to lambda e to the power negative lambda y, where y is greater than 0 and 0 or else.
00:59
Then, by the property of the exponential function, we know the mean value of y is equal to 1 over lambda.
01:07
So, as we know from this property, we know the parameter here should be one half.
01:18
Okay, now, not here because our y has been occupied, let's just use z here.
01:34
Suppose z is an exponential distribution, so we have this property.
01:44
Now let's define i to be equal to the sub -mention from i to 10.
01:52
We want to find the moment generating function, i mean expectation of e to the power of t y.
01:59
By the independence of those xi of those samples, this is equal to the product from 1 to 10 expectation of t xi and they are iid.
02:22
So we know this is equal to the expectation of t x1 to the power 10.
02:30
So we only need to find the homogeneity function for the single sample x1.
02:38
Okay by the definition this is equal to the integral from zero to positive infinity e to the power of tx times the density for x1 which is equal to one half times e to the power of negative one half x dx.
02:56
Okay, without this constant out we have 1 half times the integral from 0 to the positive infinity e to the power t minus 1 half x.
03:17
So by the property of the exponential function, we know this integral is convergent for t less than 1 half.
03:26
If t is greater than 1 half, then we have some exponentially increasing function, then the whole integral would be divergent.
03:35
So i mean if t is less than one half, then we have some exponential decay function.
03:43
Then this function is the exponential decay.
03:46
So the integral is convergent.
03:52
Now under this assumption, we know this can be written as 1 over 2 power x d t e minus one half.
04:10
Okay, so we have 1 half times 1 over t minus 1 half e to the power, evaluate at positive infinity, 0.
04:24
So as i said, t is less than 1 half, so when x goes to positive infinity this term will just go away...