Two streams flow into a reservoir. Let X and Y be two continuous random variables representing the flow of each stream with joint pdf: f(x, y) = cx y(1 + y) for 0 ≤ x ≤ 3 and 0 ≤ y ≤ 3, f(x, y) = 0 otherwise. The value c, a constant, is approximately equal to:

c = 0.165

Step-by-step explanation:

Given:

f(x, y) = cx y(1 + y) for 0 ≤ x ≤ 3 and 0 ≤ y ≤ 3,

f(x, y) = 0 otherwise.

Required:

The value of c

To find the value of c, we make use of the property of a joint probability distribution function which states that

where a and b represent -infinity to +infinity (in other words, the bound of the distribution)

By substituting cx y(1 + y) for f(x, y)  and replacing a and b with their respective values, we have

Since c is a constant, we can bring it out of the integral sign; to give us

Open the bracket

Integrate with respect to y

Substitute 0 and 3 for y

Add fraction

Rewrite;

The is a constant, so it can be removed from the integral sign to give

Integrate with respect to x

Substitute 0 and 3 for x

Multiply both sides by