The Erik Jonsson School of Engineering and Computer Science Chapter 2 pp. 49-100 William J. Pervin...
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Transcript of The Erik Jonsson School of Engineering and Computer Science Chapter 2 pp. 49-100 William J. Pervin...
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
The Erik Jonsson School of Engineering and Computer Science
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.
The Erik Jonsson School of Engineering and Computer Science
Chapter 2
Shorthand notation:
{X=x} ≡ {s S | X(s) = x}
Discrete vs. Continuous RVs
The Erik Jonsson School of Engineering and Computer Science
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}]
The Erik Jonsson School of Engineering and Computer Science
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)
The Erik Jonsson School of Engineering and Computer Science
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)
The Erik Jonsson School of Engineering and Computer Science
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)
The Erik Jonsson School of Engineering and Computer Science
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
The Erik Jonsson School of Engineering and Computer Science
Chapter 2
Poisson (α) RV: (α > 0)
PX(x) = αxe-α /x! for x=0,1,…; 0 otherwise
(Arrivals: α = λT)
The Erik Jonsson School of Engineering and Computer Science
Chapter 2
2.4 Cumulative Distribution Function
The CDF (FX) of a random variable X is
FX(x) = P[X ≤ x]
The Erik Jonsson School of Engineering and Computer Science
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.
The Erik Jonsson School of Engineering and Computer Science
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)
The Erik Jonsson School of Engineering and Computer Science
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
The Erik Jonsson School of Engineering and Computer Science
Chapter 2
Note:
Poisson PMF is limiting case of binomial PMF.
The Erik Jonsson School of Engineering and Computer Science
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)
The Erik Jonsson School of Engineering and Computer Science
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
The Erik Jonsson School of Engineering and Computer Science
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
The Erik Jonsson School of Engineering and Computer Science
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]
The Erik Jonsson School of Engineering and Computer Science
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