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90 Cards in this Set

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  • 3rd side (hint)
Product rule:

(fg)' = ?
Product rule:

(fg)' = f'g + fg'
Derivative Rules
Quotient rule:

(f / g)' = ?
Quotient rule:

(f / g)' = (f'g − fg') / (g^2)
Derivative Rules
Chain rule:

(h(g(x)))' = ?
Chain rule:

(h(g(x)))' = h'(g(x))g'(x)
Derivative Rules
Exponent derivative:

(e^x)' = ?
Exponent derivative:

(e^x)' = e^x
Derivative Rules
Fraction derivative:

(1/x^c)' = ?
Fraction derivative:

(1/x^c)' = -c / (x^c+1)
Derivative Rules
Sqrt derivative:

√x = ?
Sqrt derivative:

√x = 1 / 2√x
Derivative Rules
Binomial P(y):

P(y) = ?

Range: ?
Binomial P(y):

P(y) = (n C y)(p^y)(q^n-y)

Range: 0 <= y <= n
Discrete Distributions
Binomial E(y)&V(y):

E(y) = ?
V(y) = ?
Binomial E(y)&V(y):

E(y) = np
V(y) = npq
E(Y)&V(Y)
Binomial m(t):

m(t;n,p) = ?
Binomial m(t):

m(t;n,p) = (pe^t + q)^n
moment generating functions
Geometric P(y):

P(y) = ?

Range: ?
Geometric P(y):

P(y) = pq^y-1

Range: y > 1
Discrete Distributions
Geometric E(y)&V(y):

E(y) = ?
V(y) = ?
Geometric E(y)&V(y):

E(y) = 1 / p
V(y) = q / p^2
E(Y)&V(Y)
Geometric m(t):

m(t;p) = ?
Geometric m(t):

m(t;p) = pe^t /
1-qe^t
moment generating functions
Negative Binomial P(y):

P(y) = ?

Range: ?
Negative Binomial P(y):

P(y) = (y-1 C r-1)(p^r)(q^y-r)

Range: y >= r
Discrete Distributions
Negative Binomial E(y)&V(y):

E(y) = ?
V(y) = ?
Negative Binomial E(y)&V(y):

E(y;r,p) = r / p
V(y;r,p) = rq / p^2
E(Y)&V(Y)
Negative Binomial m(t):

m(t;r,p) = ?
y >= r
Negative Binomial m(t):

m(t;r,p) = (pe^t / 1-qe^t)^r
y >= r
moment generating functions
Poisson P(y):

P(y) = ?

Range: ?
Poisson P(y):

P(y) = (λ^y)(e^-λ) / y!

Range: y >= 0
Discrete Distributions
Poisson E(y)&V(y):

E(y) = ?
V(y) = ?
Poisson E(y)&V(y):

E(y) = λ
V(y) = λ
E(Y)&V(Y)
Poisson m(t):

m(t;λ) = ?
Poisson m(t):

m(t;λ) = e^λ(e^t - 1)
moment generating functions
Hypergeometric P(y):

P(y) = ?

Range: ?
Hypergeometric P(y):

P(y) = (r C y)(N-r C n-y) / (N C n)

Range: 0 <= Y <= min(n,r)
Discrete Distributions
Hypergeometric E(y)&V(y):

E(y) = ?
V(y) = ?
Hypergeometric E(y)&V(y):

E(y;r,n,N) = nr / N
V(y;r,n,N) = n(r/N)((N-r)/N)((N-n)/(N-1))
E(Y)&V(Y)
Variance V(y):

V(y) = ?
Variance V(y):

V(y) = E(Y^2) - E(Y)^2
Discrete Distributions
Чебышёв's Theorem:

P(|?|<?) ≤ ?

P(|?|>=?) >= ?
Чебышёв's Theorem:

P(|Y-μ|<kσ) ≤ 1 - 1/(k^2)

P(|Y-μ|≥kσ) ≥ 1/(k^2)
Discrete Distributions
Binomial Expansion:

Σ(n C i)(x^n-1)(y^i) = ?
[i=0,n]
Binomial Expansion:

Σ(n C i)(x^n-1)(y^i) = (x + y)^n
[i=0,n]
Identities
Geometric Series 1:

Σr^i = ?
[i=0,∞]
r < 1
Geometric Series 1:

Σr^i = 1 / 1-r
[i=0,∞]
r < 1
Identities
Geometric Series 2:

Σr^i = ?
[i=1,∞]
r < 1
Geometric Series 2:

Σr^i = r / 1-r
[i=1,∞]
r < 1
Identities
Geometric Series 3:

Σr^i = ?
[i=0,m]
r < 1
Geometric Series 3:

Σr^i = 1-r^(m+1) / 1-r
[i=0,m]
r < 1
Identities
Taylor Series:

Σ x^i / i! = ?
[i=0,∞]
Taylor Series:

Σ x^i / i! = e^x
[i=0,∞]
Identities
Sums 1:

Σi = ?
[i=1,n]
Sums 1:

Σi = n(n+1) / 2
[i=1,n]
Identities
Sums 2:

Σi^2 = ?
[i=1,n]
Sums 2:

Σi^2 = n(n+1)(2n+1) / 6
[i=1,n]
Identities
Sums 3:

Σi^3 = ?
[i=1,n]
Sums 2:

Σi^3 = (n(n+1) / 2)^2
[i=1,n]
Identities
Total outcomes:

total outcomes = ?
Total outcomes:

total outcomes = mn
Counting Rules
Arrangement

# ways to arrange n items = ?
Arrangement

# ways to arrange n items = n!
Counting Rules
Permutations
# possibilities when order matters

P = ?
Permutations
# possibilities when order matters

P = n!/(n-r)!
Counting Rules
Selections
# possibilities when order does not matter

(n C r) = ?
Selections
# possibilities when order does not matter

(n C r) = n!/(r!(n-r)!)
Counting Rules
Multinomial
# possibilities when order does not matter

(n C n1 n2 .. nk) = ?
Multinomial
# possibilities when order does not matter

(n C n1 n2 .. nk) = n!/(n1!n2!..nk!)
Counting Rules
Union:

P(A or B) = ?
Union:

P(A or B) = P(A) + P(B) - P(A and B)
Probability
Conditional Probability:

P(A|B) = ?
Conditional Probability:

P(A|B) = P(A and B) / P(B)
Probability
Independence:

P(A and B) = ?
Independence:

P(A and B) = P(A)P(B)
Probability
Bayes' Theorem:

P(Ai|B) = ?
Bayes' Theorem:

P(Ai|B) = P(B|Ai)P(Ai) / Σ[j]P(B|Aj)P(Aj)
Probability
Total Probability:

P(A) = ?
Total Probability:

P(A) = P(A and B) + P(A and B')
Probability
Binomial Defn:

Probability of?
Binomial Defn:

Probability of Y successes with p probability in sample n.
Discrete Distributions
Geometric Defn:

Probability of?
Geometric Defn:

Probability of Y trials before first success with probability p.
Discrete Distributions
Poisson Defn:

Probability of?
Poisson Defn:

Probability of Y incidents with λ rate of arrival.
Discrete Distributions
Negative Binomial (Pascal) Defn:

Probability of?
Negative Binomial (Pascal) Defn:

Probability of Y trials before rth success with probability p.
Discrete Distributions
Hypergeometric Defn:

Probability of?
Hypergeometric Defn:

Probability of selecting Y successes without replacement in a sample n from population N in which there are r total success.
Discrete Distributions
Properties of a Distribution Function:
1. ?
2. ?
3. ?
Properties of a Distribution Function:
1. F(-∞) = 0
2. F(∞) = 1
3. F(y1) <= F(y2) for any y1<y2
Continuous Distributions
Properties of a Density Function:
1. ?
2. ?
Properties of a Density Function:
1. f(y) >= 0
2. ʃ[-∞,∞]f(y)dy = 1
Continuous Distributions
Probability Intervals:

P(a≤Y≤b) = ?
Probability Intervals:

P(a≤Y<=b) = ʃ[a,b]f(y)dy
Continuous Distributions
Expected Value E(Y):

E(Y) = ?
Expected Value:

E(Y) = ʃ[-∞,∞] yf(y)dy
E(Y) = Σ[i=0,∞] yP(y)
E(Y)&V(Y)
Uniform f(y):

f(y) = ?

Range: ?
Uniform Distribution:

f(y) = 1/(Θ2-Θ1)

Range: Θ1 <= y <= Θ2
Continuous Distribution
Uniform E(y)&V(y):

E(y) = ?
V(y) = ?
Uniform E(y)&V(y):

E(y) = (Θ2+Θ1)/2
V(y) = (Θ2-Θ1)^2/12
E(Y)&V(Y)
E(aY+b) = ?
E(aY+b) = aE(Y) + b
Continuous Distributions
V(aY+b) = ?
V(aY+b) = (a^2)V(Y)
Continuous Distributions
Derivative Rules:

1. ?
2. ?
3. ?
4. ?
5. ?
Derivative Rules:

1. Product Rule
2. Quotient Rule
3. Chain Rule
4. Exponent Derivative
5. Fraction Derivative
Lists
Discrete Distributions:

1. ?
2. ?
3. ?
4. ?
5. ?
Discrete Distributions:

1. Binomial
2. Geometric
3. Poisson
4. Hypergeometric
5. Negative Binomial (Pascal)
Lists
Characteristic Functions:

1. ?
2. ?
3. ?
4. ?
Characteristic Functions:

1. P(Y) = ?
2. E(Y) = ?
3. V(Y) = ?
4. m(t) = ?
Lists
Identities:

1. ?
2. ?
3. ?
4. ?
Identities:

1. Binomial Expansion
2. Geometric Series
3. Taylor Series
4. Increasing Sums
Lists
Probability Theorems:

1. ?
2. ?
3. ?
4. ?
5. ?
Probability Theorems:

1. Union
2. Conditional Probability
3. Bayes Theorem
4. Total Probability
5. Independence
Lists
Moment Generating Function m(t):

m(t) = ?
Moment Generating Function m(t):

m(t) = E(e^ty) = 1 + Σ μi't^i / i!
m(t) = ʃe^ty P(y)dy
[-∞,∞]
moment generating functions
Normal f(y):

f(y) = ?
Normal f(y):

f(y) = e^(-(y-μ)^2/(2σ^2))/ σ√2π
Continuous Distributions
Normal E(y)&V(y):

E(y) = ?
V(y) = ?
Normal E(y)&V(y):

E(y) = μ
V(y) = σ^2
E(Y)&V(Y)
Normal m(t):

m(t) = ?
Normal m(t):

m(t) = e^(μt+((σt)^2)/2)
moment generating functions
Gamma f(y):

f(y) = ?

range = ?
Gamma f(y):

f(y) = y^(α-1) e^(-y/β) / β^αΓ(α)

range = y > 0
Continuous Distributions
Gamma Γ(α):

Γ(α) = ?
Gamma Γ(α):

Γ(α) = Int y^(α-1)e^-y dy
[0,∞]

Γ(α) = (α-1)!, α is integer
Continuous Distributions
Gamma E(y)&V(y):

E(y) = ?
V(y) = ?
Gamma E(y)&V(y):

E(y) = αβ
V(y) = αβ^2
E(Y)&V(Y)
Gamma m(t):

m(t) = ?
Gamma m(t):

m(t) = (1 - βt)^-α
moment generating functions
Beta f(y):

f(y) = ?

Range: ?
Beta f(y):

f(y) = y^(α-1)(1-y)^(β-1) / Β(α,β)

Range: 0 < y < 1
Continuous Distributions
Beta Β(α,β):

Β(α,β) = ?
Beta Β(α,β):

Β(α,β) = Int[0,1] y^α-1(1-y)^β-1 dy
Β(α,β) = Γ(α)Γ(β)/Γ(α+β)
Continuous Distributions
Beta E(y)&V(y):

E(y) = ?
V(y) = ?
Beta E(y)&V(y):

E(y) = α/(α+β)
V(y) = αβ/((α+β)^2(α+β+1))
E(Y)&V(Y)
Exponential α,β:

α = ?
β = ?
Exponential α,β:

α = 1
β = β
Continuous Distributions
Χ^2 α,β:

α = ?
β = ?
Χ^2 α,β:

α = v/2
β = 2
Continuous Distributions
Continuous Distributions:

1. ?
2. ?
3. ?
4. ?
5. ?
6. ?
7. ?
Continuous Distributions:

1. Uniform
2. Normal
3. Standard Normal
4. Gamma
5. Exponential
6. Chi-Square
7. Beta
Lists
Standard Normal μ,σ,z:

μ = ?
σ = ?
z = ?
Standard Normal μ,σ,z:

μ = 0
σ = 1
z = (y-μ)/σ
Continuous Distributions
Mat451 Subjects:

1. ?
2. ?
3. ?
4. ?
5. ?
6. ?
Mat451 Subjects:

1. Discrete Distributions
2. Continuous Distributions
3. Characteristic Functions
4. Probability Theorems
5. Derivative Rules
6. Identities
Lists
Uniform Example:

Example = ?
Uniform Example:

Example = The time of a single event within with a Poisson interval
A consequence of the Probability Integral Transform is that random numbers can be used for the purpose of simulating a large number of classical continuous or discrete distributions.
Continuous Distributions
Normal Example:

Example = ?
Normal Example:

Example = Amounted of liquid dispensed by bottling machine
Continuous Distributions
Gamma Example:

Example = ?
Gamma Example:

Example = Length of time between malfunctions for aircraft engines
Continuous Distributions
Exponential Definition:

Example = ?
Exponential Definition:

Definition = The probability that a component will continue to operate for b more time given that it has already operated a>0 time is the same as if a=0
"Memoryless property"
Consider a very large number of identical radioactive atoms, and observe their decay.
Continuous Distributions
Beta Example:

Example = ?
Beta Example:

Example = The proportion of impurities in a chemical product
Continuous Distributions
Chi-Square E(y)&V(y):

E(y) = ?
V(y) = ?
α = ?
β = ?
Chi-Square E(y)&V(y):

E(y) = ν
V(y) = 2ν
α = ν/2
β = 2
E(Y)&V(Y)
Exp E(y)&V(y):

E(y) = ?
V(y) = ?
α = ?
β = ?
Exp E(y)&V(y):

E(y) = β
V(y) = β^2
α = 1
β = β
E(Y)&V(Y)
Uniform m(t):

m(t) = ?
Uniform m(t):

m(t) = (e^tΘ2 - e^tΘ1) /
t(Θ2 - Θ1)
moment generating functions
Exp m(t):

m(t) = ?
Exp m(t):

m(t) = (1 - βt)^-1
moment generating functions
Chi-Square m(t):

m(t) = ?
Chi-Square m(t):

m(t) = (1 - 2t)^-ν/2
moment generating functions
Taylor Series:

e^x = ?
Taylor Series:

e^x = Σ x^i / i!
[i=0,∞]
Identities
Binomial Expansion:

(x + y)^n = ?
Binomial Expansion:

(x + y)^n = Σ(n C i)(x^n-i)(y^i)
[i=0,n]
Identities
Series Expansion

ln(1+x) = ?
Series Expansion

ln(1+x) = x - x^2/2 + x^3/3 - x^4/4 + ...
Identities
Natural Log of Power

ln(x^2) = ?
Natural Log of Power

ln(x^2) = 2ln(x)
Identities
e^x = ?
n->∞
e^x = lim(1+x/n)^n
n->∞
Counting
Definition of e

e^x = ?
n->∞
Definition of e

e^x = lim(1-x/n)^x
n->∞
Counting