Probability

Question 1

define event

Answer 1

subset of the sample space

Question 2

the 3 kolmogorov’s axioms

Answer 2

1. P(A) >= 0 for any event A
2. P(sample space) = 1
3. if A1, A2, … are mutually exclusive then P(A1 u A2 u …) = P(A1) + P(A2) + …

Question 3

mutually exclusive rule

Answer 3

P(A u B) = P(A) + P(B)

Question 4

inclusion-exclusion principle

Answer 4

P(A u B) = P(A) + P(B) - P(A u B)

Question 5

independent rule

Answer 5

P(A n B) = P(A).P(B)

Question 6

rules with independence

Answer 6

• do not say A & B are independent if A n B = no set
• do not use rule unless it is given or you have valid justification for assuming A and B are independent
• if A and B are independent then A and ¬B are independent, then ¬A and ¬B are independent

Question 7

bayes rule

Answer 7

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

Question 8

total probability

Answer 8

P(B) = P(B n A1) + P(B n A2) + P(B n A3)
= P(B|A1).P(A1) + P(B|A2).P(A2) + P(B|A3).P(A3)

Question 9

define random variables

Answer 9

• usually denoted as X, Y, Z
• is a function from a sample space to R

Question 10

probability distribution

Answer 10

a description of the probability of all outcomes in the sample space

Question 11

2 methods to describe probability distributions

Answer 11

1. discrete
2. continuous

Question 12

examples of discrete probability distributions

Answer 12

• table
• probability mass function (PMF)
• cummulative distributive function (CMF)

Question 13

examples of continuous probability distributions

Answer 13

• probability density function
• CDF

Question 14

describe probability mass function

Answer 14

• expressed as a table or a graph
• all function values add up to 1
• fx(x) = P(X = x)

Question 15

cummulative distribution functions

Answer 15

Fx(x) = P(X <= x)

Question 16

functions of a random variable

Answer 16

if X is a discrete random variable, and g:R->R is any function, then Y = g(X) is also discrete random variable. its value is completely determined by X

Question 18

linearity of expectation

Answer 18

if x is a discrete random variable, and Y = aX + b, then R(Y) = aE(X) + b for some a, b

Question 19

law of the unconsious statistician

Answer 19

E(Y) = sum(g(x)fx(x))

Question 20

variance

Answer 20

• Var(x) or o^2x
• Var(ax + b) = a^2Var[x]

Question 21

expected value with variance

Answer 21

• E[(x - ux)^2] - ux = E(x)
• Var(x) = E[x^2] - (E[x])^2

Question 22

standard deviation formula

Answer 22

ox = root(o^2x) = root(Var(x))

Question 23

uniform distribution

Answer 23

• X ~ Uniform(x)
• if X is a discrete random variable that takes a value in finite set of consecutive integers and x takes each value with equal probability

Question 24

PMF in uniform distribution

Answer 24

P(X = x) = 1/|A|

Question 25

expected value in uniform distribution

Answer 25

n + 1/2

Question 26

bernoulli distribution

Answer 26

- X ~ Bernoulli(p) - p if x = 1 - 1 - p if x = 0

Question 27

expected value in bernoulli distribution

Answer 27

E(X) = p

Question 28

variance in bernoulli distribution

Answer 28

Var(x) = p(1-p)

Question 29

binomial distribution

Answer 29

- X ~ Binomial(n, p) - fx(x) = (n x)^x(1 - p)^(n - x) if x >= 0

Question 30

expected value in binomial distribution

Answer 30

E(x) = np

Question 31

variance in binomial distribution

Answer 31

np(1 - p)