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Problem 2. (23 marks). Let X, Y, Z be i.i.d. continuous random variables with common distribution function F and common density f. Define the discrete random variables U := 1 (X ≤ Y) and V := 1 (X ≤ Z). Recall that, for all n; m ∈ N, then x=0 is F $$ \int_0^1 t^m (1-t)^n dt = \int_0^1 t^n (1-t)^m dt = \frac{n!m!}{(m+n+1)!}$$ You may use this fact freely in the sequel. $$ \int_0^1 P(X \le Y | X = y) f(y) dy $$ 1. (10 marks). Show that U follows a Bernoulli distribution of probability 1/2. In other words, show that U takes values in {0,1} and that $$ P(U = 1) = P(U = 0) = \frac{1}{2}. $$ Hint: Start by using the fact that $$ P(U = 1) = \int_R P(U = 1 | Y = y) f(y) dy. $$ At one point, it may be useful to perform the change of variables w = F(y). 2. (2 marks). Show that V follows a Bernoulli distribution of probability 1/2. In other words, show that V takes values in {0,1} and that $$ P(V = 1) = P(V = 0) = \frac{1}{2}. $$ 3. (1 mark). Deduce that E (U) = E (V) = 1/2 and that Var (U) = Var (V) = 1/4.

          Problem 2. (23 marks). Let X, Y, Z be i.i.d. continuous random variables with common distribution function F and common density f. Define the discrete random variables
U := 1 (X ≤ Y)
and
V := 1 (X ≤ Z).
Recall that, for all n; m ∈ N,
then x=0 is F
$$ \int_0^1 t^m (1-t)^n dt = \int_0^1 t^n (1-t)^m dt = \frac{n!m!}{(m+n+1)!}$$
You may use this fact freely in the sequel.
$$ \int_0^1 P(X \le Y | X = y) f(y) dy $$
1. (10 marks). Show that U follows a Bernoulli distribution of probability 1/2. In other words, show that U takes values in {0,1} and that
$$ P(U = 1) = P(U = 0) = \frac{1}{2}. $$
Hint: Start by using the fact that
$$ P(U = 1) = \int_R P(U = 1 | Y = y) f(y) dy. $$
At one point, it may be useful to perform the change of variables w = F(y).
2. (2 marks). Show that V follows a Bernoulli distribution of probability 1/2. In other words, show that V takes values in {0,1} and that
$$ P(V = 1) = P(V = 0) = \frac{1}{2}. $$
3. (1 mark). Deduce that E (U) = E (V) = 1/2 and that Var (U) = Var (V) = 1/4.

problem 2 23 marks let x y z be iid continuous random variables with common distribution function f and common density f define the discrete random variables u 1 x y and v 1 x z recall that 41981

Added by Michael I.

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