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
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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])
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)© BrainMass Inc. brainmass.com September 29, 2022, 1:36 pm ad1c9bdddf>