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### 90 Cards in this Set

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 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(|?|=?) >= ? Чебышёв's Theorem: P(|Y-μ|= 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