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Chapter 2 pp. 49-100

Chapter 2 pp. 49-100. William J. Pervin The University of Texas at Dallas Richardson, Texas 75083. Chapter 2. Discrete Random Variables. Chapter 2. 2.1 Definitions :

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Chapter 2 pp. 49-100

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  1. Chapter 2pp. 49-100 William J. Pervin The University of Texas at Dallas Richardson, Texas 75083

  2. Chapter 2 Discrete Random Variables

  3. Chapter 2 2.1 Definitions: A random variable (X) consists of a experiment with a probability measure P[.] defined on a sample space S and a function X that assigns a real number X(s) to each outcome sS.

  4. Chapter 2 Shorthand notation: {X=x} ≡ {s  S | X(s) = x} Discrete vs. Continuous RVs

  5. Chapter 2 2.2 Probability Mass Function: The PMF (PX) of the discrete random variable X is PX(x) = P[X=x] = P[{s  S | X(s) = x}]

  6. Chapter 2 Theorem: For any discrete random variable X with PMF PX and range SX: • (x) PX (x) ≥ 0 • ΣxSX PX(x) = 1 • (BSX) P[X  B] = P[B] = ΣxB PX(x)

  7. Chapter 2 2.3 Families of Discrete RVs Bernoulli (p) RV: (0 < p < 1) PX(x) = 1-p if x=0, p if x=1, 0 otherwise (Two outcomes)

  8. Chapter 2 Geometric (p) RV: (0 < p < 1) PX(x) = (1-p)x-1p, x=1,2,…; 0 otherwise (Number to first success) Binomial (n,p) RV: (0 < p < 1; n = 1,2,…) PX(x) = C(n,x)px(1-p)n-x (Number of successes in n trials) (Note: Binomial(1,p) is Bernoulli)

  9. Chapter 2 Pascal (n,p) RV: (0 < p < 1; n = 1,2,…) PX(x) = C(x-1,n-1)pk(1-p)x-n (Number to n successes) (Note: Pascal(1,p) is Geometric) Discrete Uniform (m,n) RV: (m<n integers) PX(x) = 1/(n-m+1) for x=m,m+1,…,n; 0 otherwise

  10. Chapter 2 Poisson (α) RV: (α > 0) PX(x) = αxe-α /x! for x=0,1,…; 0 otherwise (Arrivals: α = λT)

  11. Chapter 2 2.4 Cumulative Distribution Function The CDF (FX) of a random variable X is FX(x) = P[X ≤ x]

  12. Chapter 2 For any discrete random variable X with range SX = {x1 ≤ x2 ≤ …}: the CDF (FX) is monotone non-decreasing from 0 to 1, with jump discontinuities of height PX(xi) at each xi SX and constant between the jumps.

  13. Chapter 2 FX(b) – FX(a) = P[a < X ≤ b]

  14. Chapter 2 2.5 Averages Statistics: mean, median, mode, … Parameter of a model: mode, median Expected Value of X = E[X] = μX = ΣxSX xPX(x)

  15. Chapter 2 E[X] = p if X is Bernoulli (p) RV E[X] = 1/p if X is geometric (p) RV E[X] = α if X is Poisson (α) RV E[X] = np if X is binomial (n,p) RV E[X] = k/p if X is Pascal (k,p) RV E[X] = (m+n)/2 if X is discrete uniform (m,n) RV

  16. Chapter 2 Note: Poisson PMF is limiting case of binomial PMF.

  17. Chapter 2 2.6 Functions of a Random Variable Derived RV Y = g(X) for RVs when y = g(x) for values PY(y) = Σx:g(x)=y PX(x)

  18. Chapter 2 2.7 Expected Value of a Derived RV If Y = g(X) then E[Y] = μY = ΣxSX g(x)PX(x) For any RV X: E[X-μX] = 0 and E[aX + b] = aE[X] + b

  19. Chapter 2 E[X2] = ΣxSX x2 P(x)

  20. Chapter 2 2.8 Variance and Standard Deviation Var[X] = E[(X-μX)2] σX = sqrt(Var[X]) Var[X] = E[X2] – (E[X])2= E[X2] – μX2

  21. Chapter 2 Moments of a RV X: nth moment: E[Xn] nth central moment: E[(x – μX)n] Theorem: Var[aX + b] = a2 Var[X]

  22. Chapter 2 Var[X] = p(1-p) if X is Bernoulli (p) RV Var[X] = (1-p)/p2 if X is geometric (p) RV Var[X] = α if X is Poisson (α) RV Var[X] = np(1-p) if X is binomial (n,p) RV Var[X] = k(1-p)/p2 if X is Pascal (k,p) RV Var[X] = (n-m)(n-m+2)/12 if X is discrete uniform (m,n) RV

  23. Chapter 2 2.9 Conditional PMF PX|B(x) = P[X=x|B] 2.10 MATLAB

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