00:01
For this problem, to begin, we'll note that since we have that y is equal to x squared, we know that squared values are always only going to be between zero and positive infinity.
00:12
So we have that the range of y is just from zero inclusive up to infinity.
00:20
So we have then that the cumulative distribution function for y, we know that that is equal to the probability that big y is less than or equal to little y, which is then equal to the probability that x to the power of 2 is less than or equal to little y, which equivalently we can write as the probability that negative root y is less than or equal to x less than or equal to positive root y.
00:50
Because we have x squared, actually i'm sort of skipping a step here, but it'll fill that step in.
00:57
So if we take the square root of both sides of that inequality, we have probably, or just focusing on the x squared less than or equal to y part, if we take the square root of both sides, square root of x squared is the absolute value of x.
01:13
So we know that the absolute value of x must be, oops, must be less than or equal to the square root of y, which then means that x itself must be between plus or minus square root of y.
01:25
So now we can use the probability density function of x to find that probability.
01:32
It will be the integral from negative y up to positive y, or pardon me, negative root y up to positive root y, of 1 over 2, e to the power of the magnitude of x d x...