Download Discrete Random Variables

Discrete Random Variables
Reading: Chapter 2.1, 2.2, and 2.3
Homework: 2.2.3, 2.2.9, 2.3.1, 2.3.4, 2.3.6
Random Variable
A random variable, X, consists of (1) an
experiment with a probability measure
defined on the sample space S and (2) a
function that maps each outcome s in S to a
real number X(s).
The range of the r.v. is SX = {xЄR|X(s)=x for some sЄS}.
For a given x, set (event) {sЄS|X(s)=x} is written as {X=x}.
Examples:
Roll a die: S={1,2,3,4,5,6}; X: the number facing up (same as
the outcome); SX={1,2,3,4,5,6}; {X=4} = {4}
Roll two dice: S={11, 12, …, 66}; X: sum of the two numbers
facing up; SX={2,3,4,5,6,7,8,9,10,11,12}; {X=4}={13,22,31}.
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Discrete Random Variable
X is a finite random variable if its range SX is
a finite set.
X is a discrete random variable if its range SX
is a countable set.
Example: monitor the incoming phone call
Type of the call: S={V, D, F}. X: X(V) = 0, X(D)=1,
X(F) = 2; SX={0,1,2}. X: discrete and finite.
Length of the call: S={positive real numbers} = SX.
X not discrete.
Length of the call (in seconds): S={positive
integers} = SX. X: discrete, but not finite.
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Example 2.7
A free throw is good (g) or bad (b)
with equal probability. X is the
number of points from two free
throws.
S={gg, gb, bg, bb}
SX={0,1,2}
{X=0} = {bb}, {X=1}={gb,bg}, {X=2}={gg}
P[X=0]=1/4, P[X=1]=1/2, P[X=2]=1/4
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Problem 2.2.5
Y is the number of points from the
1 + 1 shooting in college basketball.
Each free throw is good (g) with
probability p.
S={b, gg, gb}
SY={0,1,2}
{Y=0} = {b}, {Y=1}={gb}, {Y=2}={gg}
P[Y=0]=1-p, P[Y=1]=p(1-p), P[Y=2]=p2
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Probability Mass Function
PMF of a discrete random variable X:
PX(x) = P[X=x]
Formula representation
Bar chart representation
Example 2.7
P[X=0]=1/4, P[X=1]=1/2, P[X=2]=1/4
Problem 2.2.5
P[Y=0]=1-p, P[Y=1]=p(1-p), P[Y=2]=p2
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Theorem 2.1
For a discrete random variable X with
range SX and PMF PX(x):
PX(x) ≥ 0
∑xЄSx PX(x) = 1
P[B] = ∑xЄB PX(x) for any subset B of SX
Quiz 2.2: PN(n) = c/n for n=1,2,3; 0 o.w.
What is c?
P[N=1], P[N ≥ 2], P[N > 3]
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Discrete Random Variables
Bernoulli
Geometric
Binomial
Pascal
Uniform
Poisson
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