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SUMMARY AND QUESTIONS
- Elementary Set Theory, Axiomatic Approach
− Set operations, union, intersection, complement, subtraction, − probability space, independence, mutually exclusive, distributive law, De-Morgan law, − Total probability theorem, Pr(B) = ∑ni=1 Pr(B|Ai) Pr(Ai); Bayes rule, Pr(Ai|B) = Pr(B Pr(|Ai) Pr(B) Ai)= ∑^ niPr(=1 Pr(B|ABi|) Pr(Ai) Pr(Ai)Ai). − Conditional Probability, Independence, Application of Bernoulli Trials; Examples: error correction coding, system reliability, random ICs, false-positive puzzles, circuit switches.
EE/STAT 322, #13 1
SUMMARY AND QUESTIONS (CONT.)
- Random Variable – Distribution Functions, PDF, Mean Values and Moments; − Gaussian, mean and variance, related PDFs. X ∼ N (μ, σ^2 ), relation between FX (x), Φ(x) and Q(x). F (X) = Φ(X σ− μ), Q(−x) = 1 − Q(x), and Q(x) = 1 − Φ(x). − Rayleigh, Exponential RVs, Other PDFs, uniform, quantization error and MSE.
− One PDF to another PDF. From fX (x) → fY (y), when y = g(x). fY (y) = (^) |g′^1 (x)|f (g−^1 (y)). Example: f (θ) → f (x), x = sin θ. − Conditional distributions/PDFs/mean. F (x|M ), E(x|M ), f (y|x), F (y|x). M = {X ≤ m}. Example: X ∼ N ( ¯X, σ^2 ), M = {X ≤ X¯}.
EE/STAT 322, #13 3
SUMMARY AND QUESTIONS (CONT.)
− Joint Probability distribution, joint PDF, Conditional Probability, − Bayes’ Rule, Estimation Based on Observation. − Statistical Independence, Correlation Between Random Variables, − Variance of Sum X + Y , PDF of Z = X + Y , PDF of Z = g(X, Y ), g(X, Y ) = XY. Jacobian factor J.
- Characteristic Function (CHF)
− Obtain CHF from PDF. Obtain CHF for Z = X 1 + X 2 + X 3. − Obtain moments from CHF. E(Xn), E(XiY k).
Example 3 Two RVs X and Y are independent. X have a PDF of fX (x) =
{ (^) 2(1 − x) 0 ≤ x ≤ 1 0 elsewhere Y is uniformly distributed between − 1 and 1.
(a) Find the PDF of the RV Z = X + Y ;
(b) Find the probability that 0 < Z ≤ 1.
