probability - courses.cs.washington.edu€¦ · probability 3.1 discrete random variables basics...
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Probability3.1 Discrete Random Variables Basics
Anna KarlinMost slides by Alex Tsun
Agenda● Intro to Discrete Random Variables● Probability Mass Functions● Cumulative Distribution function● Expectation
Flipping two coins
Random Variable
20 balls numbered 1..20● Draw a subset of 3 uniformly at random.● Let X = maximum of the numbers on the 3 balls.
Isupport91 111a 203
b 20
c 18
d F
Random Variable
Identify those RVs
a
bc
d
Which cont Whichhas Range 42,3a a
b bc 4d d
Random Picture
Flipping two coins
Flipping two coins
i
Flipping two coins
Probability Mass Function (PMF)
Probability Mass Function (PMF)
20 balls numbered 1..20● Draw a subset of 3 uniformly at random.● Let X = maximum of the numbers on the 3 balls.
Probability Mass Function (PMF)
20 balls numbered 1..20● Draw a subset of 3 uniformly at random.● Let X = maximum of the numbers on the 3 balls.
● Pr (X = 20)
● Pr (X = 18)
a Kaka
b Ypgc Mad
ag
Cumulative distribution function(CDF)The cumulative distribution function (CDF) of a random variable specifies for each possible real number , the probability that , that is
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Homeworks of 3 students returned randomly● Each permutation equally likely● X: # people who get their own homework
Prob Outcome w X(w)
1/6 1 2 3 3
1/6 1 3 2 1
1/6 2 1 3 1
1/6 2 3 1 0
1/6 3 1 2 0
1/6 3 2 1 1
I pmf
43ko
g L's
Probability
Alex TsunJoshua Fan
Flipping two coins
Expectation
Homeworks of 3 students returned randomly● Each permutation equally likely● X: # people who get their own homework● What is E(X)?
Prob Outcome w X(w)
1/6 1 2 3 3
1/6 1 3 2 1
1/6 2 1 3 1
1/6 2 3 1 0
1/6 3 1 2 0
1/6 3 2 1 1
X nXHP wX123P123 X 132P 132 t X 2B P 2B tX321 P321 X231 P231 t X 312 P 312
Flip a biased coin until get heads (flips independent)With probability p of coming up heads Keep flipping until the first Heads observed.
Let X be the number of flips until done.
● Pr(X = 1)
● Pr(X = 2)
● Pr(X = k)
a pkb fl p
k
c tp pd p u p
Flip a biased coin until get heads (flips independent)With probability p of coming up heads Keep flipping until the first Heads observed.
Let X be the number of flips until done. What is E(X)?
A
x'Ik o
osx at
Flip a biased coin independently Probability p of coming up heads, n coin flipsX: number of Heads observed.
● Pr(X = k)
a K pb pkapy
k
c E a pjd 2 pkap'T
Repeated coin flippingFlip a biased coin with probability p of coming up Heads n times. Each flip independent of all others.
X is number of Heads.
What is E(X)? A 20 p
a I
b 4
c 5
d 10
Repeated coin flippingFlip a biased coin with probability p of coming up Heads n times.
X is number of Heads.
What is E(X)?
Flip a Fair coin independently ● Probability p of coming up heads, n coin flips● X: number of Heads observed.
Probability3.2 More on Expectation
Alex Tsun
Agenda● Linearity of Expectation (LoE)● Law of the Unconscious Statistician (Lotus)
Linearity of Expectation (Idea)Let’s say you and your friend sell fish for a living.● Every day you catch X fish, with E[X] = 3.● Every day your friend catches Y fish, with E[Y] = 7.
how many fish do the two of you bring in (Z = X + Y) on an average day? E[Z] = E[X + Y] = e[X] + E[Y] = 3 + 7 = 10
You can sell each fish for $5 at a store, but you need to pay $20 in rent. How much profit do you expect to make? E[5Z - 20] = 5E[Z] - 20 = 5 x 10 - 20 = 30
Linearity of Expectation (Idea)Let’s say you and your friend sell fish for a living.● Every day you catch X fish, with E[X] = 3.● Every day your friend catches Y fish, with E[Y] = 7.
how many fish do the two of you bring in (Z = X + Y) on an average day? E[Z] = E[X + Y] =
You can sell each fish for $5 at a store, but you need to pay $20 in rent. How much profit do you expect to make? E[5Z - 20] = 5E[Z] - 20 = 5 x 10 - 20 = 30
Linearity of Expectation (Idea)Let’s say you and your friend sell fish for a living.● Every day you catch X fish, with E[X] = 3.● Every day your friend catches Y fish, with E[Y] = 7.
how many fish do the two of you bring in (Z = X + Y) on an average day? E[Z] = E[X + Y] = e[X] + E[Y] = 3 + 7 = 10
You can sell each fish for $5 at a store, but you need to pay $20 in rent. How much profit do you expect to make? E[5Z - 20] = 5E[Z] - 20 = 5 x 10 - 20 = 30
Linearity of Expectation (Idea)Let’s say you and your friend sell fish for a living.● Every day you catch X fish, with E[X] = 3.● Every day your friend catches Y fish, with E[Y] = 7.
how many fish do the two of you bring in (Z = X + Y) on an average day? E[Z] = E[X + Y] = e[X] + E[Y] = 3 + 7 = 10
You can sell each fish for $5 at a store, but you need to pay $20 in rent. How much profit do you expect to make? E[5Z - 20] = 5E[Z] - 20 = 5 x 10 - 20 = 30
Linearity of Expectation (LoE)
Linearity of Expectation (Proof)