CSE 312, 2011 Winter, W.L.Ruzzo

7. continuous random variables

continuous random variables

Discrete random variable: takes values in a finite or countable set, e.g.

X ∈ {1,2, ..., 6} with equal probabilityX is positive integer i with probability 2-i

Continuous random variable: takes values in an uncountable set, e.g.

X is the weight of a random person (a real number)X is a randomly selected point inside a unit squareX is the waiting time until the next packet arrives at the server

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f(x) : the probability density function (or simply “density”)

P(X < a) = F(x): the cumulative distribution function (or simply “distribution”)

P(a < X < b) = F(b) - F(a)

A key relationship:

b

pdf and cdf

3

f(x)

F(a) = ∫ f(x) dxa−∞ a

f(x) = F(x), since F(a) = ∫ f(x) dx,a−∞

ddx

Need ∫ f(x) dx (= F(+∞)) = 1-∞+∞

Densities are not probabilities; e.g. may be > 1

P(x = a) = P(a ≤ X ≤ a) = F(a)-F(a) = 0

P(a ≤ X ≤ a+ε) = F(a+ε) - F(a) ≈ ε • f(a)

I.e., the probability that a continuous random variable falls at a specified point is zeroThe probability that it falls near that point is proportional to the density

densities

4

sums and integrals; expectation

5

example

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properties of expectation

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still true, just as for discrete

just as for discrete, but w/integral

(e.g., see slides 25-27)

^

variance

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example

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continuous random variables: summary

Continuous random variable X has density f(x), and

α0.0 0.5 1.0 1.5

0.0

0.5

1.0

1.5

2.0

The Uniform Density Function Uni(0.5,1.0)

x

f(x)

uniform random variable

X ~ Uni(α,β) is uniform in [α,β]

β

0.0 0.5 1.0 1.5

0.0

0.5

1.0

1.5

2.0

The Uniform Density Function Uni(0.5,1.0)

x

f(x)

uniform random variable

X ~ Uni(α,β) is uniform in [α,β]

if α≤a≤b≤β:

X ~ Uni(α,β) is uniform in [α,β]

You want to read a disk sector from a 7200rpm disk drive. Let T be the time you wait, in milliseconds, after the disk head is positioned over the correct track, until the desired sector rotates under the head.

T ~ Uni(0, 8.33)

uniform random variable: example

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-1 0 1 2 3 4

0.0

0.5

1.0

1.5

2.0

The Exponential Density Function

x

f(x)

! = 2

! = 1

exponential random variable

X ~ Exp(λ)

exponential random variable

X ~ Exp(λ)

Memorylessness:

=1-F(t)

Examples

Radioactive decay: How long until the next alpha particle?Customers: how long until the next customer/packet arrives at the checkout stand/server?Buses: How long until the next northbound #71 bus arrives on the Ave?

Yes, they have a schedule, but given the vagaries of traffic, riders with-bikes-and-baby-carriages, etc., can they stick to it?

Gambler’s fallacy: “I’m due for a win”Relation to the Poisson: same process, different measures.

Poisson says how many events in a fixed time

Exponential says how long until the next event16

-3 -2 -1 0 1 2 3

0.0

0.1

0.2

0.3

0.4

0.5

The Standard Normal Density Function

x

f(x)

µ = 0

! = 1

normal random variable

X is a normal (aka Gaussian) random variable X ~ N(μ, σ2)

-10 -5 0 5 10

0.0

0.1

0.2

0.3

0.4

0.5

0

µ = 0

! = 1

-10 -5 0 5 10

0.0

0.1

0.2

0.3

0.4

0.5

µ = 4

! = 1

-10 -5 0 5 10

0.0

0.1

0.2

0.3

0.4

0.5

0

0 µ = 0

! = 2

-10 -5 0 5 10

0.0

0.1

0.2

0.3

0.4

0.5

0 µ = 4

! = 2

changing μ, σ

18density at μ is ≈ .399/σ

normal random variable

X is a normal random variable X ~ N(μ,σ2)

Y = aX + b E[Y] = E[aX+b] = aμ + b Var[Y] = Var[aX+b] = a2σ2

Y ~ N(aμ + b, a2σ2)

Important special case: Z = (X-μ)/σ ~ N(0,1)

Z ~ N(0,1) “standard (or unit) normal” Use Φ(z) to denote CDF, i.e.

no closed form

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Φ(.46)

-3 -2 -1 0 1 2 3

0.0

0.1

0.2

0.3

0.4

0.5

The Standard Normal Density Function

x

f(x)

µ = 0

! = 1

If Z ~N(µ,σ) what is P( µ-σ < Z < µ+σ )? P( µ - σ < Z < µ + σ ) = Φ(1) - Φ(-1) ≈ 68%

P( µ - 2σ < Z < µ + 2σ ) = Φ(2) - Φ(-2) ≈ 95% P( µ - 3σ < Z < µ + 3σ ) = Φ(3) - Φ(-3) ≈ 99%

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-3 -2 -1 0 1 2 3

0.0

0.1

0.2

0.3

0.4

0.5

The Standard Normal Density Function

x

f(x)

µ = 0

! = 1

normal approximation to binomial

X ~ Bin(n,p)E[X] = np Var[X] = np(1-p)Poisson approx. good for n large, p small (np constant)For large n: X ≈ Y ~ N(E[X], Var[X]) = N(np,np(1-p))

(p stays fixed)Normal approximation good when np(1-p) ≥ 10

DeMoivre-Laplace Theorem: Sn = number of success in n trials (with prob. p), as n→∞

normal approximation to binomial

Fair coin flipped 40 times. Probability of 20 heads?

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Exact answer:

Ross Ch5 ex 4f

Suppose X ~Unif(0,1). What is the distribution of Y = X2?

P(Y < 1/4) = P(X2 < 1/4) = P(X < sqrt(1/4)) = P(X < 1/2) = 1/2

I.e., X has half its probability density left of .5, but Y has that squished left of 1/4. Similarly, X has 0.1 probleft of 1/10, but Y pushes that left of 0.01. Generalizing, for any constant 0 < y < 1: P(Y < y) = P(X < sqrt(y)) = sqrt(y)which is the CDF FY of Y, and the PDF of Y is the derivative of that: fY(y) = 1/(2 sqrt(y)) for 0 < y < 1.

distribution of g(X)

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Ross 5.7 ex 7a

-0.2 0.2 0.6 1.0

0.00.20.40.60.81.0

x

density

-0.2 0.2 0.6 1.0

0.00.20.40.60.81.0

x

CDF

-0.2 0.2 0.6 1.00.0

1.0

2.0

3.0

y

density

-0.2 0.2 0.6 1.0

0.00.20.40.60.81.0

y

CDF

“The square of a fraction is more likely to be near

0 than 1”

fX FX

fY FY

Suppose X ~Unif(0,1). What is the distribution of Y = Xn?Generalizing again, for 0 ≤ y ≤1, the CDF is found as follows:

And taking the derivative, the density function is:

distribution of g(X)

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Ross 5.7 ex 7a

distribution of g(X)

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the central limit theorem (CLT)

Consider i.i.d. (independent, identically distributed) random vars X1, X2, X3, …

Xi has μ = E[Xi] and σ2 = Var[Xi]

As n → ∞,

Restated: As n → ∞,

How tall are you? Why?

Willie Shoemaker & Wilt Chamberlain29

Cre

dit:

Ann

ie L

eibo

vitz

, © 1

987

?

Human height is approximately normal.

Why might that be true?

R.A. Fisher (1918) noted it would follow from CLT if height were the sum of many independent random effects, e.g. many genetic factors (plus some environmental ones like diet). I.e., suggested part of mechanism by looking at shape of the curve.

in the real world…

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Male Height in Inches

Freq

uenc

y

Sixty-Four

(and hundreds more probably exist)

Meta-analysis of Dense Genecentric Association Studies Reveals Common and Uncommon Variants Associated with Height, Lanktree, et al.

The American Journal of Human Genetics 88, 6–18, January 7, 2011

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in the real world…

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in the real world…

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in the real world…

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in the real world…

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