Risk and uncertainty - formule

REPLACEMENT

FINITE DISCRETE MODELS WITH EQUALLY LIKELY OUTCOMES

Pr(E) = number of outcomes in event E = r = n(E)

M

. r objects into n urns

total number of possible outcomes n n(S) *

Multiplication : n(A) n(B) (ex. License plates, wardrobe)

Me ⑨

. ORDER

P = n!

Permutation : n r

A. n objects chosen r at a time

(n - r)!

Combination : C = n! = n

n r

M n objects chosen r at a time A

SEE

r! (n - r)! r OPERATIONS SUMMARY:

n n

n r n-r · Multiplications : when there is “and”, choose multiple things A B (elements in both)

Binomial theorem = (x+y) = ∩

x y M

r .

M · Plus : when there is “or”, choose either one or the other A B (elements in common)

∪

M

.

· Subtraction : get rid of some cases

CONDITIONAL PROBABILITY · Division : when you are counting something more times than needed (especially with sets)

Representation : In order to find the conditional probability

Pr(A∩B)

=

B)

Pr(A

M

• Pr(B)

. see Bayesian theorem

In order to find Pr(A∩B) : Morgan’s laws

Pr(A∩B) = P(A B) P(B)

M (A B) = A B

∩ ∪

In order to find Pr(B) : (A B) = A B

∪ ∩

P(B) = Pr(A∩B) + Pr(A ∩B)

M

,

Bayesian formula : causes precedes effect multiply B Pr(A B C) = Pr(A B C)Pr(B C)Pr(C)

. - ∩ ∩ ∩

M

-

Pr(A B) = P r(A∩B) = Pr(A)⋅Pr(B A) '

B

• .

Pr(A∩B) + Pr(A∩B) Pr(A)⋅Pr(B A)+Pr(A)⋅Pr(B A)

1

I i B use percentile

' y odd : middle value

B

sum where you find B ‣ formula (with 50%)

*

Baye’s Theorem median even : sum of the 2 middle value divided by 2

‣

)

B

Pr(A

·

)

Pr(B

=

)

B

Pr(A

·

)

Pr(B

=

Pr(B

=

A)

Pr(B ∩A) ‣

k

k

k

k

k

k

M n

Pr(A) )

B

Pr(A

·

)

Pr(B

Pr(A) minimum + maximum

i

i midrange

i = 1 ‣

‣

)

Pr(A∩B 2

i

✓

DISCRETE RANDOM VARIABLES measures of support probability

most frequent value

mode ⑧

‣

‣

P(X=x)=p(x) = F(x)= P(X<x) CENTRAL TENDENCY With probability function

‣

/

probability distribution cumulative distribution = E(X) = x · p(x )

‣

with µ

deals

. .

← i i

mean or expected value ( ) X .

µ = E(g(x)) = g(x · p(x )

)

µ

σ i i

X

2 2

2 ‣ variance .

V(X) = = E(X ) - (E(X))