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90 Cards in this Set
 Front
 Back
 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^ny) 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^y1 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 / 1qe^t 
moment generating functions


Negative Binomial P(y):
P(y) = ? Range: ? 
Negative Binomial P(y):
P(y) = (y1 C r1)(p^r)(q^yr) 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 / 1qe^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)(Nr C ny) / (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)((Nr)/N)((Nn)/(N1)) 
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(Yμ<kσ) ≤ 1  1/(k^2) P(Yμ≥kσ) ≥ 1/(k^2) 
Discrete Distributions


Binomial Expansion:
Σ(n C i)(x^n1)(y^i) = ? [i=0,n] 
Binomial Expansion:
Σ(n C i)(x^n1)(y^i) = (x + y)^n [i=0,n] 
Identities


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


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


Geometric Series 3:
Σr^i = ? [i=0,m] r < 1 
Geometric Series 3:
Σr^i = 1r^(m+1) / 1r [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!/(nr)! 
Counting Rules


Selections
# possibilities when order does not matter (n C r) = ? 
Selections
# possibilities when order does not matter (n C r) = n!/(r!(nr)!) 
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(AB) = ? 
Conditional Probability:
P(AB) = 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(AiB) = ? 
Bayes' Theorem:
P(AiB) = P(BAi)P(Ai) / Σ[j]P(BAj)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/(Θ2Θ1) 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) = (Θ2Θ1)^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^((yμ)^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^(α1) e^(y/β) / β^αΓ(α) range = y > 0 
Continuous Distributions


Gamma Γ(α):
Γ(α) = ? 
Gamma Γ(α):
Γ(α) = Int y^(α1)e^y dy [0,∞] Γ(α) = (α1)!, 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^(α1)(1y)^(β1) / 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^α1(1y)^β1 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. ChiSquare 7. Beta 
Lists


Standard Normal μ,σ,z:
μ = ? σ = ? z = ? 
Standard Normal μ,σ,z:
μ = 0 σ = 1 z = (yμ)/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


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


ChiSquare m(t):
m(t) = ? 
ChiSquare 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^ni)(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(1x/n)^x n>∞ 
Counting
