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90 Cards in this Set
- Front
- Back
- 3rd side (hint)
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Product rule:
(fg)' = ? |
Product rule:
(fg)' = f'g + fg' |
Derivative Rules
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Quotient rule:
(f / g)' = ? |
Quotient rule:
(f / g)' = (f'g − fg') / (g^2) |
Derivative Rules
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Chain rule:
(h(g(x)))' = ? |
Chain rule:
(h(g(x)))' = h'(g(x))g'(x) |
Derivative Rules
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Exponent derivative:
(e^x)' = ? |
Exponent derivative:
(e^x)' = e^x |
Derivative Rules
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Fraction derivative:
(1/x^c)' = ? |
Fraction derivative:
(1/x^c)' = -c / (x^c+1) |
Derivative Rules
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Sqrt derivative:
√x = ? |
Sqrt derivative:
√x = 1 / 2√x |
Derivative Rules
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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
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Binomial E(y)&V(y):
E(y) = ? V(y) = ? |
Binomial E(y)&V(y):
E(y) = np V(y) = npq |
E(Y)&V(Y)
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Binomial m(t):
m(t;n,p) = ? |
Binomial m(t):
m(t;n,p) = (pe^t + q)^n |
moment generating functions
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Geometric P(y):
P(y) = ? Range: ? |
Geometric P(y):
P(y) = pq^y-1 Range: y > 1 |
Discrete Distributions
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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)
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Geometric m(t):
m(t;p) = ? |
Geometric m(t):
m(t;p) = pe^t / 1-qe^t |
moment generating functions
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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
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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)
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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
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Poisson P(y):
P(y) = ? Range: ? |
Poisson P(y):
P(y) = (λ^y)(e^-λ) / y! Range: y >= 0 |
Discrete Distributions
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Poisson E(y)&V(y):
E(y) = ? V(y) = ? |
Poisson E(y)&V(y):
E(y) = λ V(y) = λ |
E(Y)&V(Y)
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Poisson m(t):
m(t;λ) = ? |
Poisson m(t):
m(t;λ) = e^λ(e^t - 1) |
moment generating functions
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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
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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)
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Variance V(y):
V(y) = ? |
Variance V(y):
V(y) = E(Y^2) - E(Y)^2 |
Discrete Distributions
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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
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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
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Geometric Series 1:
Σr^i = ? [i=0,∞] r < 1 |
Geometric Series 1:
Σr^i = 1 / 1-r [i=0,∞] r < 1 |
Identities
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Geometric Series 2:
Σr^i = ? [i=1,∞] r < 1 |
Geometric Series 2:
Σr^i = r / 1-r [i=1,∞] r < 1 |
Identities
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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
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Taylor Series:
Σ x^i / i! = ? [i=0,∞] |
Taylor Series:
Σ x^i / i! = e^x [i=0,∞] |
Identities
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Sums 1:
Σi = ? [i=1,n] |
Sums 1:
Σi = n(n+1) / 2 [i=1,n] |
Identities
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Sums 2:
Σi^2 = ? [i=1,n] |
Sums 2:
Σi^2 = n(n+1)(2n+1) / 6 [i=1,n] |
Identities
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Sums 3:
Σi^3 = ? [i=1,n] |
Sums 2:
Σi^3 = (n(n+1) / 2)^2 [i=1,n] |
Identities
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Total outcomes:
total outcomes = ? |
Total outcomes:
total outcomes = mn |
Counting Rules
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Arrangement
# ways to arrange n items = ? |
Arrangement
# ways to arrange n items = n! |
Counting Rules
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Permutations
# possibilities when order matters P = ? |
Permutations
# possibilities when order matters P = n!/(n-r)! |
Counting Rules
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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
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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
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Union:
P(A or B) = ? |
Union:
P(A or B) = P(A) + P(B) - P(A and B) |
Probability
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Conditional Probability:
P(A|B) = ? |
Conditional Probability:
P(A|B) = P(A and B) / P(B) |
Probability
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Independence:
P(A and B) = ? |
Independence:
P(A and B) = P(A)P(B) |
Probability
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Bayes' Theorem:
P(Ai|B) = ? |
Bayes' Theorem:
P(Ai|B) = P(B|Ai)P(Ai) / Σ[j]P(B|Aj)P(Aj) |
Probability
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Total Probability:
P(A) = ? |
Total Probability:
P(A) = P(A and B) + P(A and B') |
Probability
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Binomial Defn:
Probability of? |
Binomial Defn:
Probability of Y successes with p probability in sample n. |
Discrete Distributions
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Geometric Defn:
Probability of? |
Geometric Defn:
Probability of Y trials before first success with probability p. |
Discrete Distributions
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Poisson Defn:
Probability of? |
Poisson Defn:
Probability of Y incidents with λ rate of arrival. |
Discrete Distributions
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Negative Binomial (Pascal) Defn:
Probability of? |
Negative Binomial (Pascal) Defn:
Probability of Y trials before rth success with probability p. |
Discrete Distributions
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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
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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
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Properties of a Density Function:
1. ? 2. ? |
Properties of a Density Function:
1. f(y) >= 0 2. ʃ[-∞,∞]f(y)dy = 1 |
Continuous Distributions
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Probability Intervals:
P(a≤Y≤b) = ? |
Probability Intervals:
P(a≤Y<=b) = ʃ[a,b]f(y)dy |
Continuous Distributions
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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)
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Uniform f(y):
f(y) = ? Range: ? |
Uniform Distribution:
f(y) = 1/(Θ2-Θ1) Range: Θ1 <= y <= Θ2 |
Continuous Distribution
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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)
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E(aY+b) = ?
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E(aY+b) = aE(Y) + b
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Continuous Distributions
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V(aY+b) = ?
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V(aY+b) = (a^2)V(Y)
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Continuous Distributions
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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
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Discrete Distributions:
1. ? 2. ? 3. ? 4. ? 5. ? |
Discrete Distributions:
1. Binomial 2. Geometric 3. Poisson 4. Hypergeometric 5. Negative Binomial (Pascal) |
Lists
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Characteristic Functions:
1. ? 2. ? 3. ? 4. ? |
Characteristic Functions:
1. P(Y) = ? 2. E(Y) = ? 3. V(Y) = ? 4. m(t) = ? |
Lists
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Identities:
1. ? 2. ? 3. ? 4. ? |
Identities:
1. Binomial Expansion 2. Geometric Series 3. Taylor Series 4. Increasing Sums |
Lists
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Probability Theorems:
1. ? 2. ? 3. ? 4. ? 5. ? |
Probability Theorems:
1. Union 2. Conditional Probability 3. Bayes Theorem 4. Total Probability 5. Independence |
Lists
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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
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Normal f(y):
f(y) = ? |
Normal f(y):
f(y) = e^(-(y-μ)^2/(2σ^2))/ σ√2π |
Continuous Distributions
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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)
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Normal m(t):
m(t) = ? |
Normal m(t):
m(t) = e^(μt+((σt)^2)/2) |
moment generating functions
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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
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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
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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)
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Gamma m(t):
m(t) = ? |
Gamma m(t):
m(t) = (1 - βt)^-α |
moment generating functions
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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
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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
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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)
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Exponential α,β:
α = ? β = ? |
Exponential α,β:
α = 1 β = β |
Continuous Distributions
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Χ^2 α,β:
α = ? β = ? |
Χ^2 α,β:
α = v/2 β = 2 |
Continuous Distributions
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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
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Standard Normal μ,σ,z:
μ = ? σ = ? z = ? |
Standard Normal μ,σ,z:
μ = 0 σ = 1 z = (y-μ)/σ |
Continuous Distributions
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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
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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
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Normal Example:
Example = ? |
Normal Example:
Example = Amounted of liquid dispensed by bottling machine |
Continuous Distributions
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Gamma Example:
Example = ? |
Gamma Example:
Example = Length of time between malfunctions for aircraft engines |
Continuous Distributions
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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
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Beta Example:
Example = ? |
Beta Example:
Example = The proportion of impurities in a chemical product |
Continuous Distributions
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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)
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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)
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Uniform m(t):
m(t) = ? |
Uniform m(t):
m(t) = (e^tΘ2 - e^tΘ1) / t(Θ2 - Θ1) |
moment generating functions
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Exp m(t):
m(t) = ? |
Exp m(t):
m(t) = (1 - βt)^-1 |
moment generating functions
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Chi-Square m(t):
m(t) = ? |
Chi-Square m(t):
m(t) = (1 - 2t)^-ν/2 |
moment generating functions
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Taylor Series:
e^x = ? |
Taylor Series:
e^x = Σ x^i / i! [i=0,∞] |
Identities
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Binomial Expansion:
(x + y)^n = ? |
Binomial Expansion:
(x + y)^n = Σ(n C i)(x^n-i)(y^i) [i=0,n] |
Identities
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Series Expansion
ln(1+x) = ? |
Series Expansion
ln(1+x) = x - x^2/2 + x^3/3 - x^4/4 + ... |
Identities
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Natural Log of Power
ln(x^2) = ? |
Natural Log of Power
ln(x^2) = 2ln(x) |
Identities
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e^x = ?
n->∞ |
e^x = lim(1+x/n)^n
n->∞ |
Counting
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Definition of e
e^x = ? n->∞ |
Definition of e
e^x = lim(1-x/n)^x n->∞ |
Counting
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