ENGG 2040C: Probability Models and Applications

Andrej Bogdanov

Spring 2013

4. Random variablespart two

Review

A discrete random variable X assigns a discrete value to every outcome in the sample space.

Probability mass function of X: p(x) = P(X = x)

Expected value of X: E[X] = ∑x x p(x).

E[N]

N: number of headsin two coin flips

One die

Example from last time

F = face value of fair 6-sided dieE[F] = 1 + 2 + 3 + 4 + 5 + 6 = 3.5

1616 16 16 16 16

Two dice

S = sum of face values of two fair 6-sided diceSolution 1

2 3 4 5 6 7 89 10 11 12

spS(s) 1

36236

336

436

536

636

536

436

336

236

136

E[S] = 2 + 3 + … + 12

136236 136 =

7

We calculate the p.m.f. of S:

Two dice again

S = sum of face values of two fair 6-sided dice

F1 F2

S = F1 + F2

F1 = outcome of first dieF2 = outcome of second die

Sum of random variables

Let X, Y be two random variables.

X + Y is the random variable that assigns value X(w) + Y(w) to outcome w.

X assigns value X(w) to outcome wY assigns value Y(w) to outcome w

Sum of random variables

F1 F2 S = F1 + F2

11 1 1 212 1 2 3

21 2 1 3

… …

66 6 6 12

… …

Linearity of expectation

E[X + Y] = E[X] + E[Y]

For every two random variables X and Y

Two dice again

S = sum of face values of two fair 6-sided diceSolution 2

E[S] = E[F1] + E[F2]

= 3.5 + 3.5 = 7

F1 F2

S = F1 + F2

Balls

We draw 3 balls without replacement from this urn:

-1

1

11

-1

-1

0

What is the expected sum of values on the 3 balls?

0

-1

Balls

-1

1

11

-1

-1

00

-1S = B1 + B2

+ B3where Bi is the value of i-th ball.E[S] = E[B1] + E[B2] + E[B3]

p.m.f of B1:

-1 01

xp(x) 4

9 29

39

E[B1] = -1 (4/9) + 0 (2/9) + 1 (3/9) = -1/9same for B2, B3E[S] = 3 (-1/9) = -1/3.

Three dice

E[N] = E[I1] + E[I2] + E[I3]

Solution I1 I2 I3

N = I1 + I2 + I3

E[I1] = 1 (1/6) + 0(5/6) = 1/6E[I2], E[I3] = 1/6

= 3 (1/6)

= 1/2

Ik =1 if face value of kth die equals

0 if not

N = number of s. Find E[N].

Problem for you to solve

Five balls are chosen without replacement from an urn with 8 blue balls and 10 red balls.

What is the expected number of blue balls that are chosen?

What if the balls are chosen with replacement?

(a)

(b)

The indicator (Bernoulli) random variablePerform a trial that succeeds with probability p and fails with probability 1 – p.

01p(x) 1 – p p

x

p = 0.5

p(x)

p = 0.4

p(x)E[X] = p

If X is Bernoulli(p) then

The binomial random variable

Binomial(n, p): Perform n independent trials, each of which succeeds with probability p.

X = number of successes

ExamplesToss n coins. (# heads) is Binomial(n, ½). Toss n dice. (# s) is Binomial(n, 1/6).

A less obvious example

Toss n coins. Let C be the number of consecutive changes (HT or TH).

w C(w)

HTHHHHT 3THHHHHTHHHHHHH

20

Examples:

Then C is Binomial(n – 1, ½).

A non-example

Draw 10 cards from a 52-card deck.Let N = number of aces among the drawn cards

Is N a Binomial(10, 1/13) random variable?

No!

Different trial outcomes are not independent.

Properties of binomial random variablesIf X is Binomial(n, p), its p.m.f. is

p(k) = P(X = k) = C(n, k) pk (1 - p)n-k

We can write X = I1 + … + In, where Ii is an indicator random variable for the success of the i-th trial E[X] = E[I1] + … + E[In]

= p + … + p = np.

E[X] = np

Probability mass functionBinomial(10, 0.5) Binomial(50, 0.5)

Binomial(10, 0.3) Binomial(50, 0.3)

Investments

You have two investment choices:A: put $25 in one stock B: put $½ in each of 50 unrelated stocks Which do you prefer?

Investments

Probability model

Each stock

doubles in value with probability ½ loses all value with probability ½

Different stocks perform independently

Investments

NA = amount on choice A

NB = amount on choice B 50 × Bernoulli(½)

A: put $50 in one stock B: put $½ in each of 50 stocks

Binomial(50, ½)

E[NA] E[NB]

Variance and standard deviation

Let m = E[X] be the expected value of X.

The variance of X is the quantityVar[X] = E[(X –

m)2]The standard deviation of X is s = √Var[X]

They measure how close X and m are typically.

Calculating variancem = E[NA]

y 252q(y) 1

p.m.f of (NA – m)2

Var[NA] = E[(NA – m)2]

x 050p(x) ½½ p.m.f of NA

= 252

s = std. dev. of NA = 25

m – s m + s

Another formula for variance

Var[X] = E[(X – m)2]

= E[X2 – 2mX + m2]= E[X2] + E[–2mX] + E[m2]= E[X2] – 2m E[X] + m2

= E[X2] – 2m m + m2

= E[X2] – m2

for constant c, E[cX] = cE[X]

for constant c, E[c] = c

Var[X] = E[X2] – E[X]2

Variance of binomial random variableSuppose X is Binomial(n, p).Then X = I1 + … + In, where Ii =

1, if trial i succeeds0, if trial i fails

Var[X] = E[X2] – m2 = E[X2] – (np)2

m = E[X] = np

E[X2]= E[(I1 + … + In)2]= E[I1

2 + … + In2 + I1I2 + … + In-1In]

= E[I12] + … + E[In

2] + E[I1I2] + … + E[In-

1In]

E[Ii2] = E[Ii] = p E[Ii Ij] = P(Ii = 1 and Ij = 1)

= P(Ii = 1) P(Ij = 1) = p2

= n p = n(n-1) p2

= np + n(n-1) p2 – (np)2

Variance of binomial random variable

Suppose X is Binomial(n, p).

Var[X] = np + n(n-1) p2 – (np)2

m = E[X] = np

= np – p2 = np(1-p)

Var[X] = np(1-p)

s = √np(1-p).

The standard deviation of X is

Investments

NA = amount on choice A

NB = amount on choice B 50 × Bernoulli(½)

A: put $50 in one stock B: put $½ in each of 50 stocks

Binomial(50, ½)m m

s = 25 s = √50 ½ ½ = 3.536…

m – s m + s

Apples

About 10% of the apples on your farm are rotten.You sell 10 apples. How many are rotten?

Probability modelNumber of rotten apples you sold isBinomial(n = 10, p = 1/10).

E[N] = np = 1

Apples

You improve productivity; now only 5% apples rot.You can now sell 20 apples and only one will be rotten on average.N is now Binomial(20, 1/20).

Binomial(10, 1/10).349

.387

.194

.001

10-

10

Binomial(20, 1/20).354

.377

.189

.002

10-8 10-

26

.367

.367

.183

.003

10-7 10-

19

The Poisson random variable

A Poisson(m) random variable has this p.m.f.:

p(k) = e-m mk/k!

k = 0, 1, 2, 3, …

Poisson random variables do not occur “naturally” in the sample spaces we have seen.They approximate Binomial(n, p) random variables when m = np is fixed and n is large (so p is small)

pPoisson(m)(k) = limn → ∞ pBinomial(n, m/n)(k)

Rain is falling on your head at an average speed of 2.8 drops/second.

Divide the second evenly in n intervals of length 1/n.

Raindrops

Let Ei be the event “raindrop hits during interval i.”Assuming E1, …, En are independent, the number of drops in the second N is a Binomial(n, p) r.v.Since E[N] = 2.8, and E[N] = np, p must equal 2.8/n.

0 1

Raindrops

Number of drops N is Binomial(n, 2.8/n)

0 1

As n gets larger, the number of drops within the second “approaches” a Poisson(2.8) random variable:

Expectation and variance of Poisson

If X is Binomial(n, p) then E[X] = np Var[X] = np(1-

p)When p = m/n, we get

E[X] = m Var[X] = m(1-m/n)

As n → ∞, E[X] → m and Var[X] → m. This suggests

When X is Poisson(m), E[X] = m and Var[X] = m.

Problem for you to solve

Rain falls on you at an average rate of 3 drops/sec.

You walk for 30 sec from MTR to bus stop.

When 100 drops hit you, your hair gets wet.

What is the probability your hair got wet?

Problem for you to solve

SolutionOn average, 90 drops fall in 30 seconds.

So we model the number of drops N you receive as a Poisson(90) random variable.

Using the online Poisson calculator at or the poissonpmf(n, L) function in 13L07.py we get

P(N > 100) = 1 - ∑i = 0 P(N = i) ≈ 13.49%99