Question

Q5. Let (Ω, F , P) be a probability space. Let λ > 0 be some constant.

1. (a)  Give the definition of a random variable X on R ∪ {−∞, +∞}.

2. (b)  Define the distribution function FX for the RV X.

[6 marks]

[1 marks]

[1 marks] (c) Suppose that Y is a RV on R ∪ {−∞, +∞} with distribution function given by

􏰄3 −1e−λy (y≥0)

[6 marks]

FY (y) = 4 2 1

(y < 0).
For any a < b, determine P(Y > a), and P(Y ∈ (a, b]). [3 marks]

4. (d)  By noting {Y < b} = ∪n∈N{Y ≤ b − 1 } and using Continuity of Probability, n

   find P(Y < b). Similarly, find P(Y < ∞) and P(Y = −∞). [4 marks]

5. (e)  Is Y either a discrete RV, or a continuous RV, or a mixture of both? Explain

   your answer. [1 marks]