Let x and y be independent continuous random variables that are uniformly distributed on (0,1). let h=(x+2)y. find the probability p(lnh≥z) where z is a given number that satisfies ez< 2. your answer should be a function of z. hint: condition on x. p(lnh≥z1)= ? let x be a standard normal random variable, and let fx(x) be its cdf. consider the random variable z=fx(x). find the pdf fz(z) of z. note that fz(z) takes values in (0,1) .fz(z)= ?

Looks like there are two distinct problems here.

Conditioning on involves interpreting this probability for all possible fixed values of :

We have , and over this domain . Since is fixed, we can omit the conditioning notation to leave us with

has CDF

Then we can write our integral as

Using the method of distribution functions, we have

complete the steps to find the area of the trapezoid.

area of rectangle = 256   square units

area of triangle 1 = 24  square units

area of triangle 2 = 16  square units

area of triangle 3 =   96 square units

area of trapezoid = 120  square units

step-by-step explanation: