Chapter 2 pp. 49-100

The Erik Jonsson School of Engineering and Computer Science

Chapter 2pp. 49-100

William J. Pervin

The University of Texas at Dallas

Richardson, Texas 75083


Chapter 2

Discrete Random Variables


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.


Chapter 2

Shorthand notation:

{X=x} ≡ {s S | X(s) = x}

Discrete vs. Continuous RVs


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}]


Chapter 2

Theorem: For any discrete random variable X with PMF PX and range SX:

1. (x) PX (x) ≥ 0

2. ΣxSX PX(x) = 1

3. (BSX) P[X B] = P[B] = ΣxB PX(x)


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)


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)


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


Chapter 2

Poisson (α) RV: (α > 0)

PX(x) = αxe-α /x! for x=0,1,…; 0 otherwise

(Arrivals: α = λT)


Chapter 2

2.4 Cumulative Distribution Function

The CDF (FX) of a random variable X is

FX(x) = P[X ≤ x]


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.


Chapter 2

FX(b) – FX(a) = P[a < X ≤ b]


Chapter 2

2.5 Averages

Statistics: mean, median, mode, …

Parameter of a model: mode, median

Expected Value of X = E[X] = μX = ΣxSX xPX(x)


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


Chapter 2

Note:

Poisson PMF is limiting case of binomial PMF.


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)


Chapter 2

2.7 Expected Value of a Derived RV

If Y = g(X) then

E[Y] = μY = ΣxSX g(x)PX(x)

For any RV X: E[X-μX] = 0 and

E[aX + b] = aE[X] + b


Chapter 2

E[X2] = ΣxSX x2 P(x)


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


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]


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


Chapter 2

2.9 Conditional PMF

PX|B(x) = P[X=x|B]

2.10 MATLAB

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Chapter 2 pp. 49-100