Let X be a random variable defined by the pdf fx(x) = x[u(x) u(x ...

Transcribed Image Text:

Let X be a random variable defined by the pdf fx(x) = x²[u(x) − u(x − 1)] + ad(x - 2), where u(x) is the unit step function which is equal to 1 when x ≥ 0 and 0 otherwise, and 8(x) is the impulse function (Dirac delta function) which is the (generalized) derivative of u(x). a) Find a, E[X] and Var[X] b) Define the event W = {X≥ 0.5}. Find P[X ≤ xW] for all x € R. We shall define Fxw(x) = P[X ≤ x|W] the conditional CDF given the event W. Find a nonnegative function f: R→ R that satisfies f(t)dt = Fxw (x) Vx € R We can call this a conditional PDF given W. Find the set of all x ER over which your function f satisfies f(x) > 0. This is often called the "support" of the function. Let X be a random variable defined by the pdf fx(x) = x²[u(x) − u(x − 1)] + ad(x - 2), where u(x) is the unit step function which is equal to 1 when x ≥ 0 and 0 otherwise, and 8(x) is the impulse function (Dirac delta function) which is the (generalized) derivative of u(x). a) Find a, E[X] and Var[X] b) Define the event W = {X≥ 0.5}. Find P[X ≤ xW] for all x € R. We shall define Fxw(x) = P[X ≤ x|W] the conditional CDF given the event W. Find a nonnegative function f: R→ R that satisfies f(t)dt = Fxw (x) Vx € R We can call this a conditional PDF given W. Find the set of all x ER over which your function f satisfies f(x) > 0. This is often called the "support" of the function.

Answer rating: 100% (QA)

a The pdf o... View the full answer