Question

Suppose Brays Bayou floods every year and that the distribution of the high water mark Y has a cumulative distribution function: F(y) = 1 − 1 y 2 , y ∈ [1, [infinity]) . a. Verify that F(y) is a valid cdf. b. Find the probability density function f(y) for Y . c. Let us now suppose that the low water mark is reset at 2 instead of 1 and we use a unit of measure that is 1 5 of the original. The high water mark now becomes Z = 5(Y + 1). Find FZ(z).

Answer 1

Answer and Step-by-step explanation:

The given function is:

F(y) =1 – 1/y2 , 1 ≤ y < ∞

Verify function is valid

Limy->-∞f(y) = Limy->-∞ 0 = 0

Limy->∞f(y) = Limy->∞ 1 – 1/y2

= 1, for y ≤ 1

F(y) = 0 is constant. For y > 1, f”(y) = 2 / y3 > 0

So, function is increasing. Therefore f(y) is cdf.

Probability density function

The probability density function is

F(y) = d/dy f(y) = {(2/y2 if y>1o if y≤1 )}

z = 5(y+1)

F (z) = p (z ≤z) = p (5(y+1) ≤z)

= p(y ≤ (z/5) – 1)

F (z) ={(0 if z≤0 and 1-1/(z/5-1)2 if z>0 )