Chapter 3 Flashcards
(13 cards)
Q: What is the probability generating function (pgf)?
A: The pgf of a discrete random variable X is G(z) = E[z^X]. It helps determine the distribution and moments of X.
Q: How do you compute the expectation using pgf?
A: The expectation is E[X] = G’(1), where G’(z) is the derivative of the pgf.
Q: How do you calculate variance using pgf?
A: Variance is Var(X) = G’‘(1) + G’(1) - (G’(1))^2, using the second derivative of the pgf.
Q: What is the moment generating function (mgf)?
A: The mgf of a continuous random variable X is M(s) = E[e^(sX)]. It is used to derive moments and determine the distribution.
Q: How do you compute the expectation and variance using mgf?
A: E[X] = M’(0), and Var(X) = M’‘(0) - (M’(0))^2 using the first and second derivatives of the mgf.
Q: What is the linear transformation property of the mgf?
If Y = a + bX (where a and b are constants and X is a random variable), the Moment Generating Function (MGF) of Y is related to the MGF of X by the following formula:
M_Y(s) = E[e^{sY}] = E[e^{s(a + bX)}] = e^{as} \c M_X(bs)
MX is the mgf of X
Q: What is the joint pgf of independent random variables?
A: For independent X1, …, Xn, the joint pgf is G(z1, …, zn) = Π E[z_i^X_i], the product of the individual pgfs.
Q: What is the joint mgf of independent random variables?
A: For independent X1, …, Xn, the joint mgf is M(s1, …, sn) = Π E[e^(s_iX_i)], the product of the individual mgfs.
Q: What is the Central Limit Theorem (CLT)?
A: The CLT states that for i.i.d. variables with finite mean and variance, the sample mean approaches a normal distribution as n → ∞.
Q: How to approximate probabilities using CLT?
A: Use the formula P(a < Xn ≤ b) ≈ Φ((b - μ) * √n / σ) - Φ((a - μ) * √n / σ) for large n.
Q: How do sums of uniform random variables relate to CLT?
A: As the number of uniform variables increases, their sum approaches a normal distribution according to the CLT.
Q: What is the moment generating function (mgf) of the Gamma distribution?
A: For X ~ Gamma(α, λ), the mgf is M(s) = (1 - s/λ)^(-α), which can be used to find the distribution, expectation, and variance.
Q: What is the mgf of a normal distribution?
A: For X ~ N(μ, σ²), the mgf is M(s) = exp(μs + 0.5σ²s²), which is used to derive the expectation and variance.
