00:01
Hello everyone, so this is the question that we have.
00:03
We have to finally moment generating function, that is m g, f of the distribution defined by df is equal to half e to the bar minus modulus of x t x, where x tends from minus infinity to plus infinity.
00:35
And we have to calculate the variance as well for part a part b says that that x bd random variable is bd random variable with e of x mean of x is equal to 1 and e of x x minus 1 is equal to 4 we have to the variance of x and variance of 2 minus 3x now let's jump onto the solution of the question so if we have an mgf of x so it is given by m x of t is equal to m gf of x so we have been given e of e to the part t x from limits minus infinity to infinity to infinity to the part tx half of e to the part minus modulus x with respect to dm so that's all this what am i going to get i'm going to get minus infinity to zero e to the part tx half e to the bar minus is open the model is minus x dx plus 0 to infinity limit then e to the part t x plus half multiplied by half e to the part minus x d x so if we solve this what am i going to get half e to the part x 1 plus t divided by 1 plus t limits from minus infinity to 0 plus 0 plus half e to the part x, t minus 1, divided by t minus 1, 0 to infinity limits.
02:41
So, this is going to change to, if i find the variance of x, that is going to be by the formula, p of x squared minus e square of x.
02:54
So how do we give this, mx double dash 0 minus mx dash of 0 ,000, 0 ,000, 12 square.
03:05
So m x of t is going to be 1 divided by 1 minus t squared.
03:11
And if we take out the derivative, so m x dash of t is 2t divided by 1 minus t squared.
03:19
And how do we find the double derivative? m double derivative x of t is going to be 2 multiplied by 1, multiplied by 1 minus t squared plus 2 t squared divided by 1 minus t squared.
03:36
Through the bar.
03:40
Now put m x dash is equal to 0.
03:47
So an m double dash of 0.
03:51
This is going to be 2 multiplied by 1 plus 0 divided by 1 that is 2.
03:55
So my variance of x would be, variance of x would be equal to m h double dash of 0 minus m x dash of 0 called scale...