Flashcards
Probability
Cards in this deck (5)
- 1Chapter 2 - Univariate Discrete Random Variables What is the probability mass function (PMF) defined as? What is the sum of all px(x) equal to? What is the cumulative distribution function (CDF) defined as? State two comments about the CDF?px(x) = Pr(X=x)
1
Fx(x)=Pr(X - 2Chapter 2 - Expectation E[g(X)]=? What are the 2 special cases relating to this result?SIGMA[g(x)Pr(X=x)
1. If g(X)=X then E(X) = SIGMA[xPr(X=x)] which is the mean of X
2. If g(X)=[(X-E(X))^2] then E[(X-E(X))^2] = SIGMA[(X-E(X))^2Pr(X=x)] which is the variance of X - 3Chapter 2 - Expectation E(aX+b)=? Var(aX+b)=? If X and Y are discrete random variables then E(X+Y)=? If X and Y are independent discrete random variables then Var(X+Y)=? If Xi, i=1,...,n are independent discrete random variables then: 1. E[SIGMA(aiXi)]=? 2. Var[SIGMA(aiXi)]=?aE(X)+b
a^2Var(X)
E(X) + E(Y)
Var(X) + Var(Y)
1. SIGMA[aiE(Xi)]
2. SIGMA[ai^2Var(Xi)] - 4Chapter 2 - Probability Generating Functions Suppose that X is discrete and can only take integer, non-negative values then the probability generating function (PGF) is defined as? State 5 conditions relating to the PGF State the proof E(X) and Var(X) of the PGFGx(s) = E[s^x] = SIGMA[s^xPr(X=x)]
1. G'x(1) = E(X)
2. G''x(1) + G'x(1) - [G'x(1)]^2 = Var(X)
3. Gx(s) is non-decreasing over s>0
4. Gx(0) = Pr(X=0)
5. Gx(1) = 1
Answer to proof on separate sheet - 5Chapter Two - Uniqueness of PGF's State the 2 ways we can obtain Pr(X=k) If Gx(s)=Gy(s) what does this mean?1. Expanding Gx(s) in a power series in s and setting Pr(X=k) to the coefficient of s^k
2. Differentiating Gx(s) k times with respect to s and setting s=0. Then Pr(X=k) will equal [{Gx^(k)(0)}/k!]
That X and Y have the same distribution
Information
- Description
- Flashcards on Probability, created by Cara Wales on 27/01/2016.
- Created by
- Cara Wales
- Views
- 3
- Rating