STAT330 Study Guide - Midterm Guide: Random Variable, Exponential Distribution

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redsnail503

9 Oct 2015

School

University of Waterloo

Department

Statistics

Course

STAT330

Professor

Javid Ali

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Document Summary

Midterm 2 review: suppose x u n if (0, 1) and y u n if (0, 1) independently. V = ( 2 log x)1/2 sin(2 y ): let x n (0, 1) and y n (0, 1) independently. Let w = x and z = y. X . (a) find the joint p. d. f. of w and z. (b) find the marginal d. f. of z: let x1, . De ne yn = n i=1xi, where n z+. (a) derive the m. g. f. of ( log(x1), log(x2), . Find the limiting distribution of xn as n: let x be an exponentially distributed random variable with a mean of 1. You are given that the density function is f (x) = e x and the cumulative function is f (x) = 1 e x, x > 0. X (a) show that xn (b) show that xn x.

Related Questions

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)

[6 marks]

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]

amaranthwalrus338

10. Which of the following is a one-dimensional subspace of R3? A) {(2, y, z)?|x2 – y + 2z=0}. B) {(x,y, 2)7|x + 24 +32 =0}. C) The solution set of Av = 0 where A= 1 -2 3 -1 4 0 0 2 3 D) {(x, y, z)?|x + 2y + 32 = 4}. E) The solution set of Av = 0 where A= 1 0 2 0 2 0 11 0 5

7. The minimum value taken by the function f(x) = 3 sin(I) – 2 cos(x) on [0, 1] is (a) —2, (b) -4, (c) 5, (d) –12, (e) –10.

STAT330 Study Guide - Midterm Guide: Random Variable, Exponential Distribution

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