probability - courses.cs.washington.edu€¦ · probability 3.1 discrete random variables basics...

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)