# Conditional Density Function

Find the conditional density functions for the following experiments.

a) a number x is chosen at random in the interval [0,1], given that x>1/4

b) two numbers x and y are chosen at random in the interval [0,1], given that x>y

https://brainmass.com/statistics/density-estimation/conditional-density-function-432404

## SOLUTION This solution is **FREE** courtesy of BrainMass!

a) If we are given that x > 1/4, then we know that x must follow a uniform [1/4, 1] distribution. Hence,

f(x｜x > 1/4) = 1/(1-1/4) = 4/3 (for x in [1/4, 1])

b)

Without knowing that x > y, we are given that (x,y) are jointly distributed in the [0,1] x [0,1] square, which implies that the unconditional density is f(x,y) = 1 for (x,y) in the [0,1] x [0,1] square.

If we are given that x > y, then our domain for (x,y) gets reduced to a triangle, bounded by the line y = x and two sides of the original square. Our domain is now 1/2 the size of what it used to be, yet the density is still uniform over this smaller domain. Thus, after renormalization, our density is equal to 2 over the triangle. Formally,

f(x,y | x > y) = 2 (for (x,y) in the [0,1] x [0,1] AND x > y)

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