Statistical InferenceWeeks 1 & 2: Probability and Distribution

Types of Variables

All Variables

Categorical May be represented by

numbers, but does not make sense to add, subtract, average, etc

Numerical Makes sense to add,

subtract, average, etc(i.e., perform math operations)

Discrete Are counted and can

only take on non-negative whole numbers

Continuous Are measured and

can take on any real number (i.e., have decimal places)

Categorical Have no inherent

ordering (e.g., single, married, divorced)

Ordinal Have ordered levels

(e.g., primary, secondary, JC, university, etc)

Probability

P(A) = Probability of event A happening0 ≤ P(A) ≤ 1

Disjoint (mutually exclusive) events Cannot happen at the same time

− A card drawn from a deck cannot be both spades and hearts

− P(Spade & Heart) = 0

Non-disjoint events Can happen at the same time

− A card drawn from a deck can be both a spade and an ace

− P(Spade & Ace) = 1/52

Spade SpadeHeart Ace

Disjoint and non-disjoint events

Union of disjoint events−Probability of drawing a

Spade or a Heart from a deck of cards

P(Spade or Heart)

= P(Spade) + P(Heart)

= 13/52 + 13/52

= 26/52

Union of non-disjoint events−Probability of drawing a

Spade or an Ace from a deck of cards

P(Spade or Ace)

= P(Spade) + P(Ace) – P(Spade and Ace)

= 13/52 + 4/52 – 1/52

= 16/52

General Additional Rule = P(A or B) = P(A) + P(B) – P(A and B)

Marginal, Joint, and Conditional Probability

Marginal probability− Probability based on a single variable

P(Student = uses)

= 219/445

Joint Probability− Probability based on two or more

variables

P(Student = uses and Parent = uses)

= 125/445 = 0.28

Conditional Probability− Probability of one event conditional

upon another event

P(Student = use | parents = used)

= 125/210 = 0.60

Parents

Used Did not use

Total

Student

Uses 125 94 219

Does not Use

85 141 226

Total 210 235 445

Bayes’ Theorem

Bayes’ theorem− 𝑷 𝑨 𝑩) =

𝑷(𝑨 𝒂𝒏𝒅 𝑩)

𝑷 (𝑩)

Probability that the Children use given that the Parents also used𝑃 𝑐ℎ𝑖𝑙𝑑𝑟𝑒𝑛 = 𝑢𝑠𝑒 𝑝𝑎𝑟𝑒𝑛𝑡𝑠 = 𝑢𝑠𝑒𝑑)

= 𝑃(𝑐ℎ𝑖𝑙𝑑𝑟𝑒𝑛=𝑢𝑠𝑒 𝑎𝑛𝑑 𝑝𝑎𝑟𝑒𝑛𝑡𝑠=𝑢𝑠𝑒𝑑)

𝑃(𝑝𝑎𝑟𝑒𝑛𝑡𝑠=𝑢𝑠𝑒𝑑)

= 125/445

210/445

= 0.60

Parents

Used Did not use

Total

Children

Uses 125 94 219

Does not Use

85 141 226

Total 210 235 445

General Product Rule = P(A and B) = P(A|B) x P(B)

Bayes’ Theorem expanded Probability of women with

breast cancer in general population− P(breast cancer) = 0.017

Probability of true positive from mammogram− P(positive | breast cancer) = 0.78

− I.e., sensitivity

Probability of false positive from mammogram− P(positive | no breast cancer) =

0.10

− i.e., 1 - specificity

What is the probability that the patient has breast cancer given a positive mammogram? 𝑃(𝑐𝑎𝑛𝑐𝑒𝑟 | 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒)

= 𝑃 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑐𝑎𝑛𝑐𝑒𝑟) 𝑃(𝑐𝑎𝑛𝑐𝑒𝑟)

𝑃 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑐𝑎𝑛𝑐𝑒𝑟) 𝑃 𝑐𝑎𝑛𝑐𝑒𝑟 +𝑝 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑛𝑜 𝑐𝑎𝑛𝑐𝑒𝑟) 𝑃(𝑛𝑜 𝑐𝑎𝑛𝑐𝑒𝑟)

= 0.78 ∗ 0.017

0.78 ∗0.017+0.10 ∗0.983

= 0.119

Bayes’ theorem

𝑷 𝑨 𝑩) =𝑷(𝑨 𝒂𝒏𝒅 𝑩)

𝑷 (𝑩)

= 𝑷 𝑩 𝑨) 𝑷(𝑨)

𝑷 (𝑩)

= 𝑷 𝑩 𝑨) 𝑷(𝑨)

𝑷 𝑩 𝑨) 𝑷 𝑨 +𝑷 𝑩 𝑨𝒄)𝑷(𝑨𝒄)

Probability Tree

Cancer

No Cancer

P(cancer)0.017

P(no cancer)0.983

What is the probability that the patient has breast cancer given a positive mammogram?

Positive

Positive

Negative

Negative

P(positive | cancer)

0.78

P(negative | cancer)

0.22

P(positive | no cancer)

0.10

P(negative | no cancer)

0.90

P(cancer and positive)

0.017 x 0.78 = 0.01326

P(no cancer and positive)0.983 x 0.10

= 0.0983

𝑃(𝑐𝑎𝑛𝑐𝑒𝑟 | 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒)

= 𝑃(𝑐𝑎𝑛𝑐𝑒𝑟 𝑎𝑛𝑑 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 )

𝑃(𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒)

= 0.01326

0.01326+0.0983

= 0.119

Expected Mean

Expected Mean𝐸 𝑋

= E[𝑋 × 𝑝 𝑥 ] # sum of all values of x multiplied by its probability

What is the expected value of a dice roll?𝐸 𝑋

= 1 ×1

6+ 2 ×

1

6+ 3 ×

1

6+ 4 ×

1

6+ 5 ×

1

6+ 6 ×

1

6

= 3.5

Notation: 𝑥 : sample mean𝜇 : population mean

Mean

Mean𝑀𝑒𝑎𝑛

= 𝑥1+ 𝑥2+ 𝑥3+ …+ 𝑥𝑛

𝑛

What is the mean number of dots on each die face?𝑀𝑒𝑎𝑛

= 1+2+3+4+5+6

6

= 3.5

Notation: 𝑥 : sample mean𝜇 : population mean

Expected Variance

Expected Variance𝑉𝑎𝑟 𝑋

=E[(𝑋 − 𝜇)2] # sum square of difference between each value and mean

=E 𝑋2 − 𝐸[𝑋]2

What is the variance of a dice roll?

From previous slide, mean 𝐸 𝑋 = 3.5

𝐸 𝑋2 = 12 ×1

6+ 22 ×

1

6+ 32 ×

1

6+ 42 ×

1

6+ 52 ×

1

6+ 62 ×

1

6= 15.17

Var(X) = 𝐸 𝑋2 − 𝐸 𝑋 2 = 15.17 − 3.52 ≈ 2.9

Notation:𝑠2: sample variance𝜎2 : population variance

𝑠 : sample standard deviation𝜎 : population standard deviation

Population Variance

Population Variance𝜎2

= 1

𝑁Σ[(𝑥𝑖 − 𝜇)2]

What is the variance of dots on die faces?

Given 𝑥 = 3.5

𝜎2 = 1

6[ 1 − 3.5 2 + 2 − 3.5 2 + …+ 6 − 3.5 2]

≈ 2.9

Notation:𝑠2: sample variance𝜎2 : population variance

𝑠 : sample standard deviation𝜎 : population standard deviation

Sample Variance

Sample Variance𝑠2

= 1

𝑛−1Σ[(𝑥𝑖 − 𝑥)2]

Why n – 1?−A sample will always have smaller variance than the population. Thus, we

perform an “adjustment” to get a bigger variance that more closer approximates the population variance

− i.e., think of it as a “correction” used on samples

Notation:𝑠2: sample variance𝜎2 : population variance

𝑠 : sample standard deviation𝜎 : population standard deviation

Bernoulli Distribution

Where an individual trial only has two possible outcomes

Assuming a fair coin, what is the probability of it landing on heads (i.e., success)?𝑃 𝑠𝑢𝑐𝑐𝑒𝑠𝑠 = 𝑝 ℎ𝑒𝑎𝑑𝑠 1𝑝(𝑡𝑎𝑖𝑙𝑠)0 = 0.5

Assuming an unfair coin (i.e., 𝑝 ℎ𝑒𝑎𝑑𝑠 = 0.25), what is the probability of it landing on tails (i.e., failure)? 𝑃 𝑓𝑎𝑖𝑙𝑢𝑟𝑒 = 𝑝 ℎ𝑒𝑎𝑑𝑠 0𝑝(𝑡𝑎𝑖𝑙𝑠)1 = 0.75

Binomial Distribution

Probability of k successes in n trials𝑃 𝑘 𝑠𝑢𝑐𝑐𝑒𝑠𝑠𝑒𝑠 𝑖𝑛 𝑛 𝑡𝑟𝑖𝑎𝑙𝑠 = (𝑘

𝑛) 𝑝𝑘(1 − 𝑝)(𝑛−𝑘)

where (𝑘𝑛) =

𝑛!

𝑘! 𝑛−𝑘 !

Given 7 trials, how many scenarios can have 2 successes?

(27) =

7!

2!(5!)

= 7 ×6 ×5!

2 ×1×5!

= 21

If you toss the unfair coin 7 times, what’s the probability of 2 heads (i.e., successes)?

Given 𝑃 ℎ𝑒𝑎𝑑𝑠 = 0.25𝑃 𝑘 = 2 = (2

7) × 0.252 × 0.755

= 7 ×6 ×5!

2 ×1×5!× 0.252 × 0.755

= 0.311

Normal Distribution

Unimodal (only one peak) and symmetric

68-95-99.7% rule− 68% of values within 1sd from mean

− 95% of values within 2sd from mean

− 99.7% of values within 3sd from mean

Represented as 𝑁(𝜇, 𝜎)

Xiao MingMuthu

Normal Distribution

You want to compare between two cousins and determine who fared better. Xiao Ming scored 1800 on his SAT and Muthuscored 24 on his ACT—who did better?− 𝑆𝐴𝑇 𝑠𝑐𝑜𝑟𝑒𝑠 ~ 𝑁 𝑚𝑒𝑎𝑛 = 1500, 𝑆𝐷 = 300

−𝐴𝐶𝑇 𝑠𝑐𝑜𝑟𝑒𝑠 ~ 𝑁(𝑚𝑒𝑎𝑛 = 21, 𝑆𝐷 = 6)

Xiao Ming: 1800 −1500

300= 1sd

Muthu: 24 −21

6= 0.5sd

Normal Distribution (Z scores)

Standardization with Z scores (normalization)

𝑍 =𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛 − 𝜇

𝑆𝐷

Standardized (Z) score of a value is the number of standard deviations it falls above or below the mean

Z score of mean = 0

Normal Distribution

Suppose that your company ad campaign receives daily ad clicks that are (approximately) normally distributed with mean = 1,020 and standard deviation = 50. What’s the probability of getting more than 1,160 clicks a day?

𝑍 =𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛 − 𝜇

𝑆𝐷

=1,160 − 1,020

50= 2.8

𝑃 𝑍 > 2.8 = 1 − 0.9974= 0.0026

Normal Distribution

Your friend boast that his ad is in the top 25% of the company’s ad campaign. What is the lowest number of ad clicks his ad received? −𝐴𝑑 𝑐𝑙𝑖𝑐𝑘𝑠 ~ 𝑁(1020, 50)

𝑍 = 0.67 =𝑥 − 1,020

50𝑥 = 0.67 × 50 + 1020= 1053.5

Poisson Distribution

Poisson Distribution

𝑃 𝑋 =𝑒−𝜆𝜆𝑥

𝑥!− 𝑒 = 𝑏𝑎𝑠𝑒 𝑜𝑓 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑙𝑜𝑔, 2.71828…

− 𝜆 = 𝑚𝑒𝑎𝑛 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑢𝑐𝑐𝑒𝑠𝑠𝑒𝑠 𝑖𝑛 𝑎 𝑔𝑖𝑣𝑒𝑛 𝑡𝑖𝑚𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙

2.5 people show up at a bus stop every hour. What is the probability that 3 or fewer people show up after 4 hours?

𝑃 𝑋 ≤ 3 =𝑒−10100

0!+𝑒−10101

1!+𝑒−10102

2!+𝑒−10103

3!= 0.10336

Thank you for your attention!Eugene Yan