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Discrete Random Variables 2. Random Variable Numerical attribute of an experimental outcome. Probability Mass Function (PMF). Functions of Random Variables Y = 4*H 3 + 75 Y = H – E(H) Y = 1 if H = 0 0 if H >= 1. Expectation - PowerPoint PPT Presentation
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Discrete Random Variables 2
Random Variable Numerical attribute of an experimental outcome.
Probability Mass Function (PMF)
0
1/8
2/8
3/8
4/8
0 1 2 3
# of heads
pro
bab
ilit
y
p(h)
Functions of Random Variables Y = 4*H3 + 75 Y = H – E(H) Y = 1 if H = 0 0 if H >= 1
Expectation Weighted average of all possible outcomes.
E[x] = ∑ [ x px(x) ]
Variance Measures the spread of the PMF around the expected value. or Y = (X – E(X))2
σx2 = E(Y)
Functions of Random Variables (cont) Y = 1 if h <= 1 0 if h >= 2
E(H) = 1.5 E(Y) = ?
In general, for any var. X
and func. g(X): if Y = g(X): E(Y) = ∑ [ g(X) px(X) ]
Bernoulli Random Variable
Experiment: Toss coin once
0 (T) 1 (H)
XT
H
Examples of experiments with 2 possible outcomes: - is a person healthy or sick? - do you like a song on pandora.com? - will event A occur or not?
0.5 0.51-p p
P(X=1) = p
P(X=0) = 1-p
Bernoulli Random Variable (cont)
0 (T) 1 (H)
X
Experiment: Toss coin oncep
1-pPMF:
E(X) =
variance(X) =
p
p(1-p)
CDF:
0 (T) 1 (H)
X
1
1-p
Binomial Random Variable
Experiment: number of tosses: 4 probability of heads: ¾ X = number of heads
TH
TH
TH
TH
TH
TH
TH
TH
H
T
H
T
H
T
H
T
T
H
T
H
H
T
HHHHHHHT
HHTHHHTT
HTHHHTHT
HTTHHTTT
THHHTHHT
THTHTHTT
TTHHTTHT
TTTHTTTT
Generalized Experiment: number of tosses: n probability of heads: p
P(X = k) = ?
P(X=2) ?P(X=3) ?
¾
¼
¾
¼
¾
¼(n C k) pk (1-p)n-k
6 x (¾)2 x (¼)2 = 27/128 = 0.21 4 x (¾)3 x (¼) = 108/256 = 0.42
¼ ¼
¾¾
Binomial Random Variable (cont)
k
k
PMF:
CDF:
E(X) = np
Variance(X) = np(1-p)
Geometric Random Variable
Experiment: number of tosses: 3 probability of heads: ¾ X = number of tosses until you get heads
H1
T1
P(H1) = 3/4
P(T1) = 1/4
H2
T2
P(H2|T1) = 3/4
P(T2 | T1) = 1/4
P(T1 H2) = 12/64P(H1) = 48/64
H3
T3
P(H2|T1 T2) = 3/4
P(T3 | T1 T2) = 1/4
P(T1 T2 H2) = 3/64P(X=3) = ?
Generalized Experiment: number of tosses: n probability of heads: pP(X = k) = ?
Geometric Random Variable (cont)
PMF:
CDF:
E(X) =
Variance(X) =
Independence of Random Variables
Some thoughts
What does it mean for 2 experiments to be independent?
How do you derive the properties of binomial random variables from Bernoulli random variables?
Other topics: - PMFs of more than one random variable - conditional PMF
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