Exam 1 Flashcards
The ______ coefficient is used to choose k out of n objects
binomial
Equation of the binomial coefficient (n k)
n! / k! (n-k)!
If S is the sample space for any event A⊆S . . .
A ⋂ Ac = Ø, they are _________
A ⋃ Ac = S, they are _________
disjoint, partitions
Demorgan’s Law
(A ⋃ B)c = Ac ⋂ Bc
(A ⋂ B)c = Ac ⋃ Bc
P(Ø) = _____
0
P(S) = ______
1
if A⊆B, then P(A) ____ P(B)
≤
Inclusion-Exclusion Principle
P(A ⋃ B) = P(A) + P(B) - P(A ⋂ B)
If A and B are disjoint events then,
P(A ⋂ B) = _______
0
If A and B are disjoint events then,
P(A ⋃ B) = _____
P(A) = _______
P(A ⋂ B) = ____
P(A) + P(B)
1 - P(Ac)
Ø
Events A and B are independent if . . .
P(A ⋂ B) = P(A) ⋅ P(B)
P(A | B) = P(A)
P(B | A) = P(B)
Conditional Probability
P(A | B) = P(A ⋂ B)/ P(B)
When working conditionally,
P(A ⋂ B) = ________
P(A | B) ⋅ P(B) OR
P(B | A) ⋅ P(A)
Alternative formula for conditional probability
P(A | B) = [P(B | A) ⋅ P(A)] / P(B)
The Law of Total Probability for A1 …. An as partitions of S
P(B) = [P(B | A1) ⋅ P(A1)] + [P(B | A2) ⋅ P(A2)]…. up to An
What is the practical formula for Baye’s Theorem given that A and Ac are partitions of S?
P(A | B) =
P(B | A) ⋅ P(A) / P(B | A) ⋅ P(A) + P(B | Ac) ⋅ P(Ac)
Prosecutor’s Fallacy states that
P(A | B) ______ P(B | A)
does NOT equal
Baye’s Theorem
P(A | B) = [P(B | A) ⋅ P(A)] / P(B)
Disjoint events (P(A ⋂ B) = 0) can only be independent if . . .
P(A) = 0 OR P(B) = 0
What is the probability mass function (PMF) of a d.r.v?
d.r.v = X
P(X = x) for all x values in the support of random variable X
What is the cumulative distribution function (CDF) of a d.r.v?
d.r.v = X
Function F(X) = P(X ≤ x)
When describing the CDF of
P(a ≤ X ≤ b) = ________
F(b) - F(a)
The expected value of a d.r.v is a ________ average of all the possible values X can take on weighted by their ________.
weighted, probabilities
E(X) = _________
∑ xn ⋅ P(X = xn)
E(X^2) = _________
∑ xn^2 ⋅ P(X = xn)
- square each x value not probability
E(c) = _____
c
E(X + Y) = _______
E(X) + E(Y)
E(X + c) = _______
E(X) + c
E(cX) = _______
c ⋅ E(X)
Var(X) = __________
E[ (X - E(X))^2 ] OR
E(X^2) - [E(X)]^2
The standard deviation is calculated as ___________ and is used to report since its units ___________
SD = √Var(X), are NOT squared
Var(c) = ______
0
Var(X + c) = _______
Var(X) + 0
Var(c ⋅ X) = _______
c^2 ⋅ Var(X)
If X and Y are independent,
Var(X + Y) = ________
Var(X) + Var(Y)
To calculate F(X) where P(X ≤ xk)
∑ P(X = x) up to the value of xk
Basically sum all prior probabilities
For mutually exclusive events A and B,
P(A ⋂ B) = ____
0
For independent events A and B,
P(A ⋃ B) = _________
P(A) + P(B) - P(A and B)
If A and B are not disjoint then,
P(A ⋃ B) = _________
P(A) + P(B) - P(A ⋂ B)
How do you determine if events are mutually exclusive?
P(A ⋂ B) = 0
For mutually exclusive events A and B,
P(A ⋃ B) = _________
P(A) + P(B)
Mutually exclusive is also known as ____
disjoint
