3.2 More on Expectation 3.3 Variance and Standard Deviation Anna - - PowerPoint PPT Presentation

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3.2 More on Expectation 3.3 Variance and Standard Deviation Anna - - PowerPoint PPT Presentation

Jun 14, 2023 386 likes •735 views

3.2 More on Expectation 3.3 Variance and Standard Deviation Anna Karlin Most Slides by Alex Tsun Agenda Linearity of Expectation (LoE) Law of the Unconscious Statistician (Lotus) Variance Independence of random variables

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3.2 More on Expectation 3.3 Variance and Standard Deviation

Anna Karlin Most Slides by Alex Tsun

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Agenda

  • Linearity of Expectation (LoE)
  • Law of the Unconscious Statistician (Lotus)
  • Variance
  • Independence of random variables
  • Properties of variance
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Linearity of Expectation (LoE)

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Corollary: linearity for sum of lots of r.v.s

E(X1 + X2 + . . . + Xn) = E(X1) + E(X2) + . . . + E(Xn)

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Proof by induction!

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Indicator random variable

  • For any event A, can define the indicator random variable

for A

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Computing complicated expectations

  • Often boils down to finding the right way to decompose

the random variable into simple random variables (often indicator random variables) and then applying linearity

  • f expectation.
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Linearity is special!

X = ( 1 with prob 1/2 −1 with prob 1/2

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  • In general E(g(X)) 6= g(E(X))
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SLIDE 8

Linearity is special!

  • In general
  • How DO we compute ?

E(g(X)) 6= g(E(X))

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E(g(X))

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Homeworks of 3 students returned randomly

  • Each permutation equally likely
  • X: # people who get their own homework
  • What is E(X2 mod 2)?

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 1/6 3 1 2 1/6 3 2 1 1

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SLIDE 10

Law of the Unconscious Statistician (Lotus)

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SLIDE 11

Probability

Alex Tsun Joshua Fan

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Variance (Intuition)

Which game would you rather play? We flip a fair coin. Game 1:

  • If heads, You pay me $1.
  • If Tails, I pay you $1.

Game 2:

  • If Heads, you pay me $1000.
  • If Tails, I pay you $1000.
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SLIDE 13

Variance (Intuition)

how far is a random variable from its mean, on average?

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Variance (Intuition)

how far is a random variable from its mean, on average?

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SLIDE 15

Variance (Intuition)

how far is a random variable from its mean, on average?

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Variance and Standard Deviation (SD)

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Variance and Standard Deviation (SD)

More Useful

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SLIDE 18

Variance and Standard Deviation (SD)

More Useful

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SLIDE 19

Variance (Property)

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Variance (Property)

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Variance (Example)

LOTUS

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Variance

Which game would you rather play? We flip a fair coin. Game 1:

  • If heads, You pay me $1.
  • If Tails, I pay you $1.

Game 2:

  • If Heads, you pay me $1000.
  • If Tails, I pay you $1000.
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SLIDE 23

In general, var(X+Y) Var(X) + Var(Y)

Example 1:

  • each with prob ½
  • E(X)= ? Var(X) = ?
  • How about Y = -X

Example 2: What is Var(X+X)?

6=

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X = ±1

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Variance in pictures

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Random Picture

probability students Definition of Expectation

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Random variable X and event E are independent if the event E is independent of the event {X=x} (for any fixed x), i.e. ∀x P(X = x and E) = P(X=x) • P(E) Two random variables X and Y are independent if the events {X=x} and {Y=y} are independent for any fixed x, y, i.e. ∀x, y P(X = x and Y=y) = P(X=x) • P(Y=y) Intuition as before: knowing X doesn’t help you guess Y or E and vice versa.

Random variables and independence

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Example

Random variable X and event E are independent if the event E is independent of the event {X=x} (for any fixed x), i.e. ∀x P(X = x and E) = P(X=x) • P(E) Example: Let X be number of heads in n independent coin

  • flips. Let E be the event that the number of heads is even.
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Example

Two random variables X and Y are independent if the events {X=x} and {Y=y} are independent (for any fixed x, y), i.e. ∀x, y P(X = x and Y=y) = P(X=x) • P(Y=y) Example: Let X be number of heads in first n of 2n independent coin flips, Y be number in the last n flips, and let Z be the total.

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r.v.s and independence

Example continued

Example: Let X be number of heads in first n of 2n independent coin flips, Y be number in the last n flips, and let Z be the total.

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Important facts about independent random variables

Theorem: If X & Y are independent, then E[X•Y] = E[X]•E[Y] Theorem: If X and Y are independent, then Var[X + Y] = Var[X] + Var[Y] Corollary: If X1 + X2 + … + Xn are mutually independent then Var[X1 + X2 + … + Xn ] = Var[X1] + Var [X2] + … + Var[Xn]

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E[XY] for independent random variables

products of independent r.v.s

!X

Note: NOT true in general; see earlier example E[X2]≠E[X]2

independence

  • Theorem: If X & Y are independent, then E[X•Y] =

E[X]•E[Y]

  • Proof:
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Variance of a sum of independent r.v.s

variance of independent r.v.s is additive

!X (Bienaymé, 1853)

Theorem: If X and Y are independent, then Var[X + Y] = Var[X] + Var[Y] Proof:

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Independent vs dependent r.v.s

  • Dependent r.v.s can reinforce/cancel/correlate in

arbitrary ways.

  • Independent r.v.s are, well, independent.

Example: Z = X1 + X2 +…. + Xn Xi is indicator r.v. with probability 1/2 of being 1. versus W = n X1

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Probability

Alex Tsun Joshua Fan