Random variables, expectation, and variance - GitHub Pages

[Pages:31]Random variables, expectation, and variance

DSE 210

Random variables

Roll a die.

Define = X

1 if die is 3 0 otherwise

Here the sample space is = {1, 2, 3, 4, 5, 6}.

Roll dice. n

! = 1, 2 ) X = 0 ! = 3, 4, 5, 6 ) X = 1

= # of 6's X

= # of 1's before the first 6 Y

Both and are defined on the same sample space, XY

= {1, 2, 3, 4, 5, 6}n. For instance,

! = (1, 1, 1, . . . , 1, 6) ) = 1, = 1. X Yn

In general, a

is a defined on a probability space.

random variable (r.v.)

It is a mapping from to R. We'll use capital letters for r.v.'s.

The distribution of a random variable

Roll a die. Define = 1 if die is 3, otherwise = 0.

X

X

takes values in {0, 1} and has distribution: X

Pr(X = 0)

=

1 3

and

Pr(X = 1)

=

2 3.

Roll dice. Define = number of 6's.

n

X

takes values in X

{0,

1,

2,

.

.

.

,

}. n

The

distribution

of

X

is:

Pr( = ) = #(sequences with 6's) ? Pr(one such sequence)

Xk

k

=n

1 k 5 nk 66

k

Throw a dart at a dartboard of radius 1. Let be the distance to the X

center of the board.

takes values in [0, 1]. The distribution of is:

X

X

Pr( ) = 2. Xx x

Expected value, or mean

The expected value of a random variable is X

X E(X ) = x Pr(X = x).

x

Roll a die. Let be the number observed. X

E(X )

=

1

?

1 6

+

2

?

1 6

+

?

?

?

+

6

?

1 6

=

1+2

+

3+4 6

+

5+6

=

3.5

(average)

Biased coin. A coin has heads probability . Let be 1 if heads, 0 if tails. pX

E(X ) = 1 ? p + 0 ? (1

) p

=

p.

Toss a coin with bias repeatedly, until it comes up heads. p

Let be the number of tosses. X

E(X )

=

1 .

p

Pascal's wager

Pascal:

I

think

there

is

some

chance

( p

> 0)

that

God

exists.

Therefore

I

should act as if he exists.

Let = my level of suering. X

I Suppose I behave as if God exists (that is, I behave myself).

Then is some significant but finite amount, like 100 or 1000. X

I Suppose I behave as if God doesn't exists (I do whatever I want to).

If indeed God doesn't exist: = 0. X

But if God exists: X = 1 (hell).

Therefore, E(X ) = 0 ? (1

) p

+

1

?

p

=

1.

The first option is much better!

Linearity of expectation

I If you double a set of numbers, how is the average aected? It is also doubled.

I If you increase a set of numbers by 1, how much does the average change?

It also increases by 1.

I

Rule:

E(aX

+

) b

=

aE(X )

+

b

for

any

random

variable

X

and any

constants , . ab

I But here's a more surprising (and very powerful) property:

E(X

+

Y

)

=

E(X )

+

E(Y )

for

any

two

random

variables

, X

. Y

I

Likewise:

E(X + Y

+

) Z

=

E(X )

+

E(Y

)

+

E(Z ),

etc.

Linearity: examples

Roll 2 dice and let Z denote the sum. What is E(Z )?

Method 1 Distribution of :

Z

2 3 4 5 6 7 8 9 10 11 12

z

Pr( = ) Zz

1 36

2 36

3 36

4 36

5 36

6 36

5 36

4 36

3 36

2 36

1 36

Now use formula for expected value:

E(Z )

=

2

?

1 36

+

3

?

2 36

+

4

?

3 36

+

?

?

?

=

7.

Method 2 Let X1 be the first die and X2 the second die. Each of them is a single die and thus (as we saw earlier) has expected value 3.5. Since Z = X1 + X2,

E(Z ) = E(X1) + E(X2) = 3.5 + 3.5 = 7.

Toss

n

coins

of

bias

, p

and

let

X

be

the

number

of

heads.

What

is

E(X )?

Let

the

individual

coins

be

X1,

.

.

.,

. Xn

Each has value 0 or 1 and has expected value .

p

Since

X

=

X1

+ X2

+???

+, Xn

E(X

)

=

E(X1)

+

?

?

?

+

E(Xn)

=

. np

Roll a die n times, and let X be the number of 6's. What is E(X )?

Let X1 be 1 if the first roll is a 6, and 0 otherwise.

E(X1)

=

1 6.

Likewise,

define

X2,

X3,

.

.

.

,

. Xn

Since

X

=

X1

+???

+, Xn

we

have

E(X )

=

E(X1) +

? ? ? + E(Xn)

=

n 6

.

Coupon collector, again

Each cereal box has one of action figures. What is the expected k

number of boxes you need to buy in order to collect all the figures?

Suppose you've already collected 1 of the figures. Let be the time

i

Xi

to collect the next one.

Each box you buy will contain a new figure with probability

( k

( i

1))/k. Therefore,

E(Xi ) = k + 1 . ki

Total

number

of

boxes

bought

is

X

=

X1

+ X2

+???

+, Xk

so

E(X ) = E(X1) + E(X2) + ? ? ? + E(Xk )

= =

k k k

+

1

k

k

+

1 2

1+ k

+???

k +

2 1

+

?

?

?

+

k 1

ln . kk

k

Independent random variables

Random variables X , Y are independent if

Pr(X

=

x, Y

=

) y

=

Pr(X

=

x )Pr(Y

= ). y

Independent or not?

I Pick a card out of a standard deck. = suit and = number.

X

Y

Independent.

I Flip a fair coin times. = # heads and = last toss.

n

X

Y

Not independent.

I , take values { 1, 0, 1}, with the following probabilities: XY

Y -1 0 1 -1 0.4 0.16 0.24 0 0.05 0.02 0.03 X 1 0.05 0.02 0.03

Independent.

XY -1 0.8 0.5 0 0.1 0.2 1 0.1 0.3

Variance

If you had to summarize the entire distribution of a r.v. by a single X

number, you would use the mean (or median). Call it ?.

But these don't capture the

of :

spread X

Pr(x)

Pr(x)

x ?

x ?

What would be a good measure of spread? How about the average distance away from the mean: E(|X ?|)?

For convenience, take the square instead of the absolute value.

Variance:

var( ) X

=

E(X

?)2 = E(X 2)

?2,

where ? = E(X ). The variance is always 0.

Variance: example

Recall:

var( X

)

=

E(X

?)2 = E(X 2)

?2, where ? = E(X ).

Toss

a

coin

of

bias

. p

Let

X

2

{0, 1}

be

the

outcome.

E(X ) = p

E(X 2) = p

E(X

?)2 = 2 ? (1 p

) + (1 p

)2 ? = (1 p pp

) p

E(X 2)

?2 = p

2 = (1 pp

) p

This variance is highest when p = 1/2 (fair coin).

p

The standard deviation of is var( ).

X

X

It is the average amount by which diers from its mean.

X

Variance of a sum

var(X1

+???+ ) Xk

=

var(X1

)

+

?

?

?

+

var( Xk

)

if

the

Xi

are

independent.

Symmetric random walk. A drunken man sets out from a bar. At each time step, he either moves one step to the right or one step to the left, with equal probabilities. Roughly where is he after steps?

n

Let 2 { Xi

1, 1}

be

his

th i

step.

Then

E(Xi )

=

?0

and

var( ) Xi

=

?1.

His

position

after

n

steps

is

X

=

X1

+

???

+

. Xn

E(X ) = 0 var( ) =

X np stddev( ) =

Xn

He is likely to be pretty close to where he started!

Sampling

Useful variance rules:

I

var(X1

+

?

?

?

+

Xk

)

=

var(X1)

+

?

?

?

+

var( Xk

)

if

's Xi

independent.

I var( + ) = 2var( ). aX b a X

What fraction of San Diegans like sushi? Call it . p

Pick people at random and ask them. Each answers 1 (likes) or 0 n

(doesn't

like).

Call

these

values

X1,

.

.

.

,

. Xn

Your

estimate

is

then:

=

X1

+

???+ Xn .

Y

n

How accurate is this estimate?

Each has mean and variance (1 ), so

Xi

p

pp

E(Y ) = E(X1) + ? ? ? + E(Xn) = p

n

var( ) Y

=

var(X1)

+???

2

+ var( ) Xn

=

(1 p

) p

r

n

n

stddev( ) = Y

(1 p

) p

2p1

n

n

DSE 210: Probability and statistics

Winter 2018

Worksheet 4 -- Random variable, expectation, and variance

1. A die is thrown twice. Let X1 and X2 denote the outcomes, and define random variable X to be the minimum of X1 and X2. Determine the distribution of X.

2. A fair die is rolled repeatedly until a six is seen. What is the expected number of rolls?

3. On any given day, the probability it will be sunny is 0.8, the probability you will have a nice dinner is 0.25, and the probability that you will get to bed early is 0.5. Assume these three events are independent. What is the expected number of days before all three of them happen together?

4. An elevator operates in a building with 10 floors. One day, n people get into the elevator, and each of them chooses to go to a floor selected uniformly at random from 1 to 10.

(a) What is the probability that exactly one person gets out at the ith floor? Give your answer in terms of n.

(b) What is the expected number of floors in which exactly one person gets out? Hint: let Xi be 1 if exactly one person gets out on floor i, and 0 otherwise. Then use linearity of expectation.

5. You throw m balls into n bins, each independently at random. Let X be the number of balls that end up in bin 1.

(a) Let Xi be the event that the ith ball falls in bin 1. Write X as a function of the Xi. (b) What is the expected value of X?

6. There is a dormitory with n beds for n students. One night the power goes out, and because it is dark, each student gets into a bed chosen uniformly at random. What is the expected number of students who end up in their own bed?

7. In each of the following cases, say whether X and Y are independent.

(a) You randomly permute (1, 2, . . . , n). X is the number in the first position and Y is the number in the second position.

(b) You randomly pick a sentence out of Hamlet. X is the first word in the sentence and Y is the second word.

(c) You randomly pick a card from a pack of 52 cards. X is 1 if the card is a nine, and is 0 otherwise. Y is 1 if the card is a heart, and is 0 otherwise.

(d) You randomly deal a ten-card hand from a pack of 52 cards. X is 1 if the hand contains a nine, and is 0 otherwise. Y is 1 if all cards in the hand are hearts, and is 0 otherwise.

8. A die has six sides that come up with dierent probabilities:

Pr(1) = Pr(2) = Pr(3) = Pr(4) = 1/8, Pr(5) = Pr(6) = 1/4.

(a) You roll the die; let Z be the outcome. What is E(Z) and var(Z)?

4-1

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download