STAT 3025Q Chapter Notes - Chapter 4: Poisson Point Process, Poisson Distribution, Log-Normal Distribution

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raspberrygrasshopper669

18 Dec 2017

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Ch 4 continuous random variables and probability distributions. Recall that a random variable is continuous if. Possible values comprise either a single interval on the number line (ei. for some. A < b, any number x between a and b is a possible value) or a union of disjoint intervals. P(x=c)=0 for any number c that is a possible value of x. Probability distribution function (pdf) - let x be a continuous rv. Then a pdf of x is a function f(x) such that for any two numbers a and b with a b. The probability that x takes on a value in the interval [a, b] is the area above this interval and under the graph of the density function. Density curve - the graph of f(x) For f(x) to be a legitimate pdf . 0 for all x f (x) f(x)dx=area under the entire graph f(x)=1. X b = p c+ f (x)dx.

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

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?