Probability

Question 1

What is a basic way to approach forecasting?

Answer 1

Take observed (empirical) frequencies as representing the probabilities of outcomes in the future
This can be sensible with large data sets on repetitive standardised phenomena but doesn’t help with conditions of uncertainty and scant sample data

Question 2

What is an experiment?

Answer 2

An event where the outcome is unknown ex ante

Question 3

What is an event?

Answer 3

A set of possible outcomes

Question 4

What is the sample space?

Answer 4

The set of possible outcomes, denoted Ω

Question 5

What is the addition rule for probabilities?

Answer 5

P(A or B) = P(A) + P(B) - P(A and B)

Question 6

What is the multiplication rule for probabilities?

Answer 6

P(A and B) = P(A)P(B|A)

Question 7

What is the condition for mutually exclusive events?

Answer 7

P(A and B) = 0

Question 8

What is the condition for independent events?

Answer 8

P(A) = P(A|B)

Question 9

What does a combination tell you?

Answer 9

nCr is the number of ways r objects can be arranged from a set of n objects (without repetition - ordered)

Question 10

What is a probability mass function?

Answer 10

The set of numbers for a discrete variable that describes the probability of each event, with each probability being between 0 and 1 inclusive and altogether summing to 1

Question 11

What is the cdf?

Answer 11

The cumulative distribution function of a random variable X is F(x) = P(X <= x)
It is non decreasing

Question 12

How do you get from a continuous random variable’s cdf to its pdf?

Answer 12

F(x) = P(X <= x) = ∫_-inf^xf(u)du so when F is differentiable the probability density function f(x) is dF(x)/dx

Question 13

What is the probability that a value lies in some range for a continuous random variable?

Answer 13

P(a <= x <= b) = ∫_a^bf(u)du

Question 14

What is the Bayesian system?

Answer 14

Priors (initial beliefs) + results of an experiment (new data) –> likelihood (improved or updated belief)

Question 15

What is the frequentist system?

Answer 15

p̂ = no. successes observed / no. events observed

Question 16

What is Bayes’ Theorem?

Answer 16

P(A|B) = P(B|A)P(A)/P(B)
P(A) is the prior
B is the observation
P(A|B) is the posterior
P(B|A) is the likelihood

Question 17

What is the sensitivity of a test?

Answer 17

P(positive test|positive event)

Question 18

What is the specificity of a test?

Answer 18

P(negative test|negative event)

Question 19

What is the false positive rate?

Answer 19

P(positive test|negative event) = 1 - specificity

Question 20

What is the false negative rate?

Answer 20

P(negative test|positive event) = 1 - sensitivity

Question 21

What is the joint probability mass function and what are the conditions on it?

Answer 21

P(x, y) = P(X = x, Y = y)
0 <= P(x, y) <= 1 for all (x, y)
Σ_iΣ_jP(x_i, y_j) = 1
If X and Y are independent then P(X = x, Y = y) = P_XY(x, y) = P(X = x)P(Y = y)

Question 22

What is a marginal probability mass function?

Answer 22

P_X(x) = P(x) = Σ_jP_XY(x, y_j)

Question 23

Can you go from the marginal probability mass functions to the joint probability mass function?

Answer 23

Not necessarily, information about the dependence relation is needed

Question 24

What is a joint probability density function?

Answer 24

P(a < X < b, c < Y < d) = ∫_a^b∫_c^df(x, y)dydx

Question 25

What is the marginal probability density function?

Answer 25

f_X(x) = ∫_-inf^inff_XY(x, y)dy

Question 26

What is the joint pdf of an IID sample?

Answer 26

Assuming X₁, X₂, … are independent and identically distributed (written X_i ~ IID), f(x₁, x₂, …) = Π_i=1^Nf(x_i) = (f(x))^N

Question 27

What is the conditional density function?

Answer 27

Found from the joint pdfs
P(x|y) = P(x, y)/P(y) with P(y) > 0 or
f_X|Y(x, y) = f_XY(x, y)/f_Y(y)
Still need to integrate to 1

Question 28

How do you go from any normal distribution to the standard normal distribution?

Answer 28

P(X < a) = P((X - μ)/σ < (a - μ)/σ) = P(Z < (a - μ)/σ)