STAT330 Lecture Notes - Sample Space

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yellowsea-lion710

9 Feb 2014

School

University of Waterloo

Department

Statistics

Course

STAT330

Professor

Christine Dupont

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Document Summary

Suppose x and y are de ned on the same sample space s: joint c. d. f. F (x, y) = 0 and lim (x,y) ( , ) : f (x, y) = p (x = x, y = y). Support set a = {(x, y) : f (x, y) > 0} (b) properties of f: f (x, y) 0 (x, y) r2 ii. (x,y) ( , ) F (x, y) = lim (cid:88) (x,y) (cid:88) Support set a = {(x, y) : f (x, y) > 0} X y (b) properties of f: f (x, y) 0 (x, y) r2 ii.

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?

brownhyena716

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

STAT330 Lecture Notes - Sample Space

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Introductory Statistics

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