Computing probabilities for random variables (Lebesgue measure)

Question

Suppose that (Ω, F, P) is Lebesgue measure on the interval [0, 1]

Define random variables X(ω)=ω, Y(ω)=2ω+3 and Z(ω)=4ω+1

Compute P(X < b ∩ Y < c) and P(Z > a) as functions of a, b, c ∈ R

Answer 1

Random variables are functions X : Ω → R so the probability of an event, which is a subset of the range of X, is equal to the measure of the preimage of the event, which is a subset of the domain of X. For example,

 

{X < b}

= {ω : X(ω) < b}

= {ω : ω < X^-1(b)}

 

In general X^-1 is a set function so it's defined even if X is not invertible, but in your case X, Y, Z happen to be invertible.

 

{X < b ∩ Y < c}

= {ω : ω < X^-1(b) and ω < Y^-1(c)}

= {ω : ω < b and ω < c/2 - 3/2}

= {ω : ω < min(b, c/2 - 3/2)}

 

so

 

P(X < b ∩ Y < c)

= μ{ω : ω < min(b, c/2 - 3/2)}

 

Since μ is uniform, the measure is the length of the overlap of [0, 1] and {ω : min(b, c/2 - 3/2)}. From here you'll have to break it up into cases depending on the values of b and c. For example, if b < 0 or c < 3 the intersection is empty so μ = 0.

 

P(Z > a)

= μ{ω : ω > a/4 - 1/4}

 

Again, find the length of the intersection with [0, 1] and consider cases depending on the value of a.