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Cards in this deck (6)
- 1Chapter 8 - Families of Continuous Random Variables For the Uniform random variable X~U(a,b) state the PDF, MGF, E(X) and Var(X) For the exponential random variable X~Exp(lambda) state the PDF, MGF, E(X) and Var(X)X~U(a,b):
fx(x) = 1/(b-a)
Mx(t) = (e^bt -e^at)/t(b-a)
E(X) = (a+b)/2 Var(X) = (b-a)^2/12
X~Exp(lambda):
fx(x) = lamba*e^(-lambdax)
Mx(t) = lambda/(lambda - t)
E(X) = 1/lambda Var(X) = 1/lambda^2 - 2Chapter 8 - MGF of a Linear Combination Suppose that X has MGF given by Mx(t). The MGF of the random variable Y=aX+b is given by? State the proof of the above statementMy(t) = e^(bt)Mx(at)
Answer on separate sheet - 3Chapter 8 - MGF of Sums of Independent random variables If X1, X2, ..., Xn are independent random variables and Y=X1+X2+...+Xn then My(t)=? State the proof of the above statementMy(t) = Mx1(t)Mx2(t)...Mxn(t)
Answer on separate sheet - 4Chapter 8 - Uniqueness of MGFs What does it mean if Mx(t) = My(t)? MGFs can also be defined for discrete random variables. For a discrete random variable X, the moment generating function (MGF) is?That X and Y have the same distribution
Mx(t) = E(e^tX) = SIGMAxe^(tx)Pr(X=x) - 5Chapter 8 - Continuity Theorem State the continuity theoremSuppose that we have
X1,X2, ... (random variables)
Fx1(x),Fx2(x), ... (CDFs)
Mx1(t),Mx2(t),... (MGFs)
If as n tends to infinity
Mxn(t) tends to Mx(t) for all t
where Mx(t) is the MGF of some random variable with CDF Fx(x), then
Fxn(x) tends to Fx(x) as n tends to infinity
at each x where Fx(x) is continuous - 6Chapter 8 - Cumulant Generating Functions If Mx(t) is the MGF of a random variable X, then the cumulant generating function is? If S=X1+X2+...+Xn then Ks(t)=?Kx(t) = log[Mx(t)] = k1t + (k2T^2)/2! + (k3t^3)/3! + ...
where:
k1 = E(X)
k2 = E[(X-E(X))^2] = Var(X)
k3 = E[(X-E(X)^3]
k4 = E[(X-E(X)^4] -3[Var(X)]^2
Ks(t) = SIGMAKxi(t)
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- Flashcards on Untitled, created by Cara Wales on 27/01/2016.
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