00:01
Given the normal distribution with parameter with mean value mu and variance sigma squared, i mean the density function for x is equal to 1 over square root of 2 pi times sigma times e to the power of the exponential function.
00:23
Negative 1 over 2 times x minus mu squared over sigma squared for x.
00:31
This is the density function for our x.
00:37
And define a new random variable y to be equal to e to the power of x.
00:42
We want to find the density function for y.
00:47
To find the density function, we want to begin the distribution function.
00:51
By the definition, the distribution for y is equal to the probability probability, y less or equal to some little y.
00:59
If there's y by our expression here, e to the power x less or equal to y.
01:06
So it's easy for us to say this will be equal to 0 when y is less or equal to 0.
01:12
Because e to some power cannot be less than, maybe it's better for us to write it in this way, cannot be less or equal to some negative thing.
01:25
Okay, when y is greater or equal to zero, then this probability is not zero.
01:36
Okay, and notice this function, logarithm for x is an increasing function for positive numbers.
01:48
So if we take the logarithm on both sides, we know the probabilities are changed.
01:59
Less or equal to the law in y and law in exponential is equal to x, x less or equal to law in y.
02:11
Okay, then use the property of the standard of the normal distribution.
02:18
This is equal to the integral from negative infinity to law in y of the density function minus mu square over sigma square dx.
02:33
Okay, then this is the distribution function for y.
02:39
So to find the density function for y, we only need to take the distribution for capital y...