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Q5. Let (Ω, F , P) be a probability space. Let λ > 0 be some constant.
(a) Give the definition of a random variable X on R ∪ {−∞, +∞}.
(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)
FY (y) = 4 2 1
(y < 0).For any a < b, determine P(Y > a), and P(Y ∈ (a, b]). [3 marks]
(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]
(e) Is Y either a discrete RV, or a continuous RV, or a mixture of both? Explain
your answer. [1 marks]
7. (10pts) (a) Find the solution of the system 3-y+32 = -1 x + 3y - 2=3 2 + y + 5z = 1 that is closest to the point P(1,-1,0) in R3.
2. (10 marks) ex-y = xy. Determine the (a) Consider the curve C implicitly defined by the equation equation of the tangent line to Cat (1,1). (b) Compute the derivative of y = (sin x)sin .