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STAT330 Study Guide - Final Guide: 2Wo, Exponential Distribution, Probability Distribution

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School
University of Waterloo
Department
Statistics
Course
STAT330
Professor
Grace Yun Yi

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Document Summary

Practice questions 2: two points are chosen uniformly and independently from [0, 1]. Find the expected distance between them: suppose x and y are continuous random variables with joint p. d. f. f (x, y) = 24xy for. Y = n x: let x be a continuous random variable with p. d. f. f (x) = 2x, 0 < x < 1. Are x and y independent? (b) find the p. d. f. of y . (c) find v ar(x|y ), the conditional variance of x given y . (d) find the p. d. f. of z = x. Practice questions 2: let x follow a uniform distribution on ( 1, 1), z n (0, 1) and is independent of x, Y = x(1 x) + z. (a) find e(y ). (b) find v ar(y ). (c) find the correlation between x and y : let x be a uniform random variable on (0, 1). 0 x > 0, otherwise, where , > 0 are constants.

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Related textbook solutions

Introductory Statistics
OER Edition, 2013
Openstax
ISBN: 9781938168208

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Related Questions

Let X and Y be continuous random variables with joint density function.

 

(a) Find the marginal densities of X and Y.

(b). Are X and Y independent?

(c). Find P (X< 1/2, Y <1)

 

 

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]

page1image62001664 page1image62003968

FY (y) = 4 2 1

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

  1. (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]

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

    your answer. [1 marks]

You are given a fair die, i.e. the outcomes of any throw are contained in the sample space   Let the event A be that an even number is thrown and the event B that a multiple of three is thrown. Now consider the two random variables X and Y. X takes on a value of 1 if the event A has occurred and the value 0 if A has not occurred. The random variable Y takes on the value of 1 if the event B has occurred and the value zero if B has not occurred.

(a) Derive the joint pdf of X and Y. Provide this in the form of a table

                                        Y=0                                Y=1

X=0    
X=1    

b) What is the marginal probability distribution of Y?
(c) What is the conditional probability distribution of X, given that Y = 1?
(d) What is E (X|Y = 1)?
(e) What is Cov (X, Y )?
(f) Are X and Y independent random variables?