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STAT330 Lecture 3: STAT 330 Lecture 3 (Spring 2017)

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School
University of Waterloo
Department
Statistics
Course
STAT330
Professor
Elaheh Torkashvand

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Document Summary

Suppose x and y are discrete random variables with joint pmf for x, y = 0, 1, 2, . and zero otherwise. Find the constant c and also the marginal pmf"s of x and y. Let x and y be continuous random variables with the joint pdf and zero otherwise. Find k such that f(x,y) is a joint pdf. Accordingly, find the marginal pdf"s of x and y. Suppose x and y are continuous random variables with the joint pdf f (x, y) = 24xy, for 0 < x < 1, y > 0, and x + y < 1, and zero otherwise. Find p(y > 2x) and the marginal pdf of x. Suppose x and y are continuous random variables with the joint pdf f (x, y) = cx4 (1 - y)3 for 0 < x <= y < 1 and zero otherwise.

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Related textbook solutions

Introductory Statistics
OER Edition, 2013
Openstax
ISBN: 9781938168208

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Related Questions

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?

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]