00:01
Okay, given a random variable x with pdf, f is equal to 4 times x to the power 3, where x is greater than 0 and less than 1.
00:16
We want to use the cdf to find the density function for y being x to the power 4 and w being x to the power x, z being 0 and u being x minus 1 half to the power 2.
00:45
Okay, now as we want to use the cdf method, it's very straightforward to, it's very, i mean, a very natural thought is to find the cdf for x first.
00:59
I mean f by the definition is the probability of x less or equal to x is equal to let's say 0 on x is less or equal to 0, 1 on x is greater or equal to 1.
01:19
Otherwise it's the integral 0 to x 4t to the power of q dt which is equal to x to the power 4.
01:32
Okay now let's consider the cdf of y.
01:36
The definition is the probability of y is equal to x to the power of 4 less or equal to y.
01:46
By the property of x, it's easy to see.
01:49
We can see x is a random variable between 0 and 1, so the probability will be equal to 0 when y is less or equal to 0, and 1 when y is less, y is greater or equal to 1.
02:05
And when y is between 0 and 1, this this set is equivalent to this guy x is less than or equal to y to the power of 1 over 4.
02:16
So by this part, we have this guy to the power of 4, which is y.
02:25
Y is greater than 0 and less than 1.
02:29
So that's the function for y is equal to 1, 0, 1.
02:36
When y is greater than 0 and less than 1, 0.
02:45
Fw, okay, it is e to the power x less than the sum of w.
02:55
Again, as x is between 0 and y, it's easy for us to see.
03:01
When w is less or equal to y, its probability is equal to 0.
03:07
When w is greater or equal to e, its probability is equal to 1.
03:11
When w is between 1 and e, this is equivalent.
03:15
And we can take the logarithm on both sides without changing anything.
03:21
X is less or equal to ln w.
03:24
So we have ln w to the power 4.
03:28
W is greater than 1 and less than e.
03:32
The density for w is the derivative of this term, which is equal to 4 times ln w to the power 3 times 1 over w.
03:43
When w is greater than 1 and less than e, it is equal to 0.
03:50
Let's say we can find the pdf for z, z is equal to 0 and 1.
04:06
Lowing x is less than 0, so it is equal to 1 when z is greater or equal to 0...