Exam 2

• Trial: Flip a coin

• Outcome: heads

• Sample space: S={heads, tails) all possible outcomes

• collection of outcomes

• 
  

(logical probability). Consider the

physical properties of a die. If all sides are equally probable,

we get

the proportion of sixes if I roll the die

an infinite number of times.

My previous experience of die rolls

and my belief about the properties of the die tells me that myprobability to roll a six is 1/6 ≈ 0.1667

• no common elements

• events with common elements

A ∩ B

• A or B occurs (at least one of them

• AUB

A^c (complement)

• when A does not occur

• 0<=P(A) <=1

• P(S) = 1 (S=Sample space)

• P(A ∩ B)= P(A) + P(B) (if A and B are disjunct)

P(A∩ B) = P(A) * P(B) (Events A and B are independent if the knowledge that B has taken place does not affect the probability of A, and vice versa)

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

n! = n(n − 1)(n − 2) · · · 2 · 1

1

• A given that B has happened

P(B|A) = (P(A ∩ B))/P(A)

• Joint: (P∩B)

• Marginal: P(A) or P(B)

• Conditional Probability: P(B|A)

P(B|A) = P(A|B)P(B)/ P(A)

Prevalence: P(Covid) - share of population that is infected

Sensitivity: P(pos|Covid) - how sensitive is the test for

detecting Covid?

Specificity: P(neg|not Covid) - is the test specific for Covid,

or is it often triggered when Covid is not present?

Spe-ci-fi-ci-ty

We write random variables with capital letters, X, Y, ... and

their numerical outcomes with lower case letters, x, y,

• Example: The random variable X is the number of dots on a die. We roll a die and get x = 3.

A random variable is discrete if you can count the possible

outcomes, even if the the number of outcomes is infinite

A random variable is continuous if the outcomes are

uncountable

• The body temperature of a patient with fever

We often use the greek letter μ to represent E[X]

The expected value is the center of the probability distribution

• The variance is often denoted by the symbol σ^2.

Normal Distribution X ∼ N(μ, σ)

• Expected Value E(X) = μ

• Standard Deviation SD(X) = σ

f(x) = 1

• The density function is used to calculate probabilities:

• P(a ≤ X ≤ b) = area under f (x) between a and b.

Z = (X − μ)/σ

• 0= μ

• 1 =sigma

• Positive covariance - the variance of the sum is greater than in the case of independence

• Negative covariance - the variance of the sum is smaller than in the case of independence

• binary outcome

• independent observations

• propability p same for each trial

Classic example: flip a coin

• 0= failure

• 1 = success

Independent Bernoulli trials, each with probability of success p, are preformed until the first success. Let X be the total number of trials. Then,

P(X = n) = (1 − p)n−1p

We say that X follows a Geometric Distribution and we denote this X ∼ Geom(p)

Easy explanation:

• Bernoulli trials until first success

• The number of trials varies - the number is the outcome

Also a number of identically distributed Bernoulli trials

• Number of trials n is predetermined

• X is the number of successes from the n trials

• Notation: X ∼ Bin(n, p)

• X follows a binomial distribution with parameters n and p

• X is the sum of n independent and identically distributed

Bernoulli trials

The density is the same everywhere where f (x) > 0

• discrete

• Used for counting

• The occurance of one event should not affect the probability of the next event

• The rate is constant

• Two events cannot occur at exactly the same time

• lambda

Maximum Likelihood - choosing the λ that maximized the

probability of the data that we have

Student-t is fat-tailed meaning that extreme outcomes are more probable than for the normal distribution

• As ν increases, it starts to resemble the normal distribution

If the population variance σ2 is known, the standardized mean follows a normal distribution.

If the population variance σ2 is unknown and must be estimated with s2. Then the standardized mean follows a Student’s t-distribution with ν = n − 1 degrees of freedom.

• to estimate population parameter

• The population mean μ is estimated using the estimator ¯ X

This kind of interval has a 95%

probability of capturing the true value every time a

proper sample is drawn

Unbiasedness means the estimator is correct on average, over all possible samples.

SD(ˆp) decreases as the sample size n increases.

• Law of Large Numbers - ˆp will be close to p in large samples.

Bias of ˆp

Bias(ˆp) = E[ˆp] − p = 0.

• Sample size n ≥ 30 (Central Limit Theorem)

• Success-Failure Condition

• np ≥ 10 and nq ≥ 10 for estimation of proportion

• Independence assumption must be (reasonably) satisfied

• The sample is at most 10% of the population

The standard deviation of a sampling distribution is called

standard error. It measures how much the statistic (e.g., the sample mean) is expected to vary from sample to sample. We often denote this SE.

• Trade-off: Higher confidence level ⇒ larger margin of error.

When we sample from a normal distribution, n ≥ 30 is not

required (we then do not need central limit theorem

H0= no change (we first assume H0 is true)

HA= change

To preform a hypothesis test

1. State the hypothesis

2. State the test statistic

3. Find the critical value. Draw!

4. Formulate the decision rule

5. Calculate the observed test statistic

6. Draw a conclusion by comparing the observed test statistic to

the decision rule

• β0 is the population intercept

• β1 is the population slope of the regression line

1. The relationship between y and x is linear

2. The error terms εi are independent

3.The error terms have a constant standard deviation

(homoskedasticity)

4. The error terms are normally distributed

Prediction interval for ˆy⋆ - two sources of uncertainty

• The unknown parameters β0 and β1

• The variation in individual y-values around the regression line. All observations are ”hit by an ε” with standard deviation σε.

Variance Inflation Factor (VIF) inflation of variance of the regression coefficient due to multicollinearity. High VIF indicates a predictor has a strong linear relationship with other predictors.