Probability 3.1 Discrete Random Variables Basics Anna Karlin Most - - PowerPoint PPT Presentation

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Probability 3.1 Discrete Random Variables Basics Anna Karlin Most - - PowerPoint PPT Presentation

Nov 26, 2023 105 likes •603 views

Probability 3.1 Discrete Random Variables Basics Anna Karlin Most slides by Alex Tsun Agenda Recap on rvs Expectation Linearity of Expectation (LoE) Law of the Unconscious Statistician (Lotus) Random Variable Probability Mass

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Probability

3.1 Discrete Random Variables Basics

Anna Karlin Most slides by Alex Tsun

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Agenda

  • Recap on rvs
  • Expectation
  • Linearity of Expectation (LoE)
  • Law of the Unconscious Statistician (Lotus)
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Random Variable

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Probability Mass Function (PMF)

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

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Expectation

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

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Repeated coin flipping

Flip 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)?

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SLIDE 9

Repeated coin flipping

Flip a biased coin with probability p of coming up Heads n times. X is number of Heads. What is E(X)?

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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

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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

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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

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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

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Linearity of Expectation (LoE)

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Linearity of Expectation (Proof)

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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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Homeworks of students returned randomly

  • Each permutation equally likely
  • X: # people who get their own homework
  • What is E(X) when there are n students?

Prob Outcome w X 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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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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Repeated coin flipping

Flip a biased coin with probability p of coming up Heads n times. X is number of Heads. What is E(X)?

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Repeated coin flipping

Flip a biased coin with probability p of coming up Heads n times. X is number of Heads. What is E(X)?

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SLIDE 22

Pairs with same birthday

  • In a class of m students, on average how many pairs of

people have the same birthday?

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Pairs with same birthday

  • In a class of m students, on average how many pairs of

people have the same birthday?

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Rotating the table

n people are sitting around a circular table. There is a nametag in each place Nobody is sitting in front of their own nametag. Rotate the table by a random number k of positions between 1 and n-1 (equally likely). X is the number of people that end up front of their own nametag. What is E(X)?

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Linearity of Expectation with Indicators

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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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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(X3 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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Law of the Unconscious Statistician (Lotus)

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Probability

Alex Tsun Joshua Fan

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Probability

3.3 Variance and standard Deviation

Most slides by Alex Tsun

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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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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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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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Variance and Standard Deviation (SD)

More Useful

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Variance (Property)

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Variance (Property)

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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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Variance (Example)

LOTUS

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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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Variance in pictures

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One more linearity of expectation practice problem Given a DNA sequence of length n e.g. AAATGAATGAATCC…… where each position is A with probability pA T with probability pT G with probability pG C with probability pC. What is the expected number of occurrences of the substring AATGAAT? AAATGAATGAATCC AAATGAATGAATCC

!X

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Random Picture

probability students Definition of Expectation