MAT 451
Terms
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-
Product rule:
(fg)' = ? -
Product rule:
(fg)' = f'g + fg' -
Quotient rule:
(f / g)' = ? -
Quotient rule:
(f / g)' = (f'g − fg') / (g^2) -
Chain rule:
(h(g(x)))' = ? -
Chain rule:
(h(g(x)))' = h'(g(x))g'(x) -
Exponent derivative:
(e^x)' = ? -
Exponent derivative:
(e^x)' = e^x -
Fraction derivative:
(1/x^c)' = ? -
Fraction derivative:
(1/x^c)' = -c / (x^c+1) -
Sqrt derivative:
√x = ? -
Sqrt derivative:
√x = 1 / 2√x -
Binomial P(y):
P(y) = ?
Range: ? -
Binomial P(y):
P(y) = (n C y)(p^y)(q^n-y)
Range: 0 <= y <= n -
Binomial E(y)&V(y):
E(y) = ?
V(y) = ? -
Binomial E(y)&V(y):
E(y) = np
V(y) = npq -
Binomial m(t):
m(t;n,p) = ? -
Binomial m(t):
m(t;n,p) = (pe^t + q)^n -
Geometric P(y):
P(y) = ?
Range: ? -
Geometric P(y):
P(y) = pq^y-1
Range: y > 1 -
Geometric E(y)&V(y):
E(y) = ?
V(y) = ? -
Geometric E(y)&V(y):
E(y) = 1 / p
V(y) = q / p^2 -
Geometric m(t):
m(t;p) = ? -
Geometric m(t):
m(t;p) = pe^t /
1-qe^t -
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 -
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 -
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 -
Poisson P(y):
P(y) = ?
Range: ? -
Poisson P(y):
P(y) = (λ^y)(e^-λ) / y!
Range: y >= 0 -
Poisson E(y)&V(y):
E(y) = ?
V(y) = ? -
Poisson E(y)&V(y):
E(y) = λ
V(y) = λ -
Poisson m(t):
m(t;λ) = ? -
Poisson m(t):
m(t;λ) = e^λ(e^t - 1) -
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) -
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)) -
Variance V(y):
V(y) = ? -
Variance V(y):
V(y) = E(Y^2) - E(Y)^2 -
Чебышёв's Theorem:
P(|?|<?) ≤ ?
P(|?|>=?) >= ? -
Чебышёв's Theorem:
P(|Y-μ|<kσ) ≤ 1 - 1/(k^2)
P(|Y-μ|≥kσ) ≥ 1/(k^2) -
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] -
Geometric Series 1:
Σr^i = ?
[i=0,∞]
r < 1 -
Geometric Series 1:
Σr^i = 1 / 1-r
[i=0,∞]
r < 1 -
Geometric Series 2:
Σr^i = ?
[i=1,∞]
r < 1 -
Geometric Series 2:
Σr^i = r / 1-r
[i=1,∞]
r < 1 -
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 -
Taylor Series:
Σ x^i / i! = ?
[i=0,∞] -
Taylor Series:
Σ x^i / i! = e^x
[i=0,∞] -
Sums 1:
Σi = ?
[i=1,n] -
Sums 1:
Σi = n(n+1) / 2
[i=1,n] -
Sums 2:
Σi^2 = ?
[i=1,n] -
Sums 2:
Σi^2 = n(n+1)(2n+1) / 6
[i=1,n] -
Sums 3:
Σi^3 = ?
[i=1,n] -
Sums 2:
Σi^3 = (n(n+1) / 2)^2
[i=1,n] -
Total outcomes:
total outcomes = ? -
Total outcomes:
total outcomes = mn -
Arrangement
# ways to arrange n items = ? -
Arrangement
# ways to arrange n items = n! -
Permutations
# possibilities when order matters
P = ? -
Permutations
# possibilities when order matters
P = n!/(n-r)! -
Selections
# possibilities when order does not matter
(n C r) = ? -
Selections
# possibilities when order does not matter
(n C r) = n!/(r!(n-r)!) -
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!) -
Union:
P(A or B) = ? -
Union:
P(A or B) = P(A) + P(B) - P(A and B) -
Conditional Probability:
P(A|B) = ? -
Conditional Probability:
P(A|B) = P(A and B) / P(B) -
Independence:
P(A and B) = ? -
Independence:
P(A and B) = P(A)P(B) -
Bayes' Theorem:
P(Ai|B) = ? -
Bayes' Theorem:
P(Ai|B) = P(B|Ai)P(Ai) / Σ[j]P(B|Aj)P(Aj) -
Total Probability:
P(A) = ? -
Total Probability:
P(A) = P(A and B) + P(A and B') -
Binomial Defn:
Probability of? -
Binomial Defn:
Probability of Y successes with p probability in sample n. -
Geometric Defn:
Probability of? -
Geometric Defn:
Probability of Y trials before first success with probability p. -
Poisson Defn:
Probability of? -
Poisson Defn:
Probability of Y incidents with λ rate of arrival. -
Negative Binomial (Pascal) Defn:
Probability of? -
Negative Binomial (Pascal) Defn:
Probability of Y trials before rth success with probability p. -
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. -
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 -
Properties of a Density Function:
1. ?
2. ? -
Properties of a Density Function:
1. f(y) >= 0
2. ʃ[-∞,∞]f(y)dy = 1 -
Probability Intervals:
P(a≤Y≤b) = ? -
Probability Intervals:
P(a≤Y<=b) = ʃ[a,b]f(y)dy -
Expected Value E(Y):
E(Y) = ? -
Expected Value:
E(Y) = ʃ[-∞,∞] yf(y)dy
E(Y) = Σ[i=0,∞] yP(y) -
Uniform f(y):
f(y) = ?
Range: ? -
Uniform Distribution:
f(y) = 1/(Θ2-Θ1)
Range: Θ1 <= y <= Θ2 -
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(aY+b) = ?
- E(aY+b) = aE(Y) + b
- V(aY+b) = ?
- V(aY+b) = (a^2)V(Y)
-
Derivative Rules:
1. ?
2. ?
3. ?
4. ?
5. ? -
Derivative Rules:
1. Product Rule
2. Quotient Rule
3. Chain Rule
4. Exponent Derivative
5. Fraction Derivative -
Discrete Distributions:
1. ?
2. ?
3. ?
4. ?
5. ? -
Discrete Distributions:
1. Binomial
2. Geometric
3. Poisson
4. Hypergeometric
5. Negative Binomial (Pascal) -
Characteristic Functions:
1. ?
2. ?
3. ?
4. ? -
Characteristic Functions:
1. P(Y) = ?
2. E(Y) = ?
3. V(Y) = ?
4. m(t) = ? -
Identities:
1. ?
2. ?
3. ?
4. ? -
Identities:
1. Binomial Expansion
2. Geometric Series
3. Taylor Series
4. Increasing Sums -
Probability Theorems:
1. ?
2. ?
3. ?
4. ?
5. ? -
Probability Theorems:
1. Union
2. Conditional Probability
3. Bayes Theorem
4. Total Probability
5. Independence -
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
[-∞,∞] -
Normal f(y):
f(y) = ? -
Normal f(y):
f(y) = e^(-(y-μ)^2/(2σ^2))/ σ√2π -
Normal E(y)&V(y):
E(y) = ?
V(y) = ? -
Normal E(y)&V(y):
E(y) = μ
V(y) = ϒ^2 -
Normal m(t):
m(t) = ? -
Normal m(t):
m(t) = e^(μt+((ϒt)^2)/2) -
Gamma f(y):
f(y) = ?
range = ? -
Gamma f(y):
f(y) = y^(α-1) e^(-y/β) / β^αΓ(α)
range = y > 0 -
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 -
Gamma E(y)&V(y):
E(y) = ?
V(y) = ? -
Gamma E(y)&V(y):
E(y) = αβ
V(y) = αβ^2 -
Gamma m(t):
m(t) = ? -
Gamma m(t):
m(t) = (1 - βt)^-α -
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 -
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) -
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)) -
Exponential α,β:
α = ?
β = ? -
Exponential α,β:
α = 1
β = β -
Χ^2 α,β:
α = ?
β = ? -
Χ^2 α,β:
α = v/2
β = 2 -
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 -
Standard Normal μ,ϒ,z:
μ = ?
ϒ = ?
z = ? -
Standard Normal μ,ϒ,z:
μ = 0
ϒ = 1
z = (y-μ)/ϒ -
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 -
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. -
Normal Example:
Example = ? -
Normal Example:
Example = Amounted of liquid dispensed by bottling machine -
Gamma Example:
Example = ? -
Gamma Example:
Example = Length of time between malfunctions for aircraft engines -
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. -
Beta Example:
Example = ? -
Beta Example:
Example = The proportion of impurities in a chemical product -
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 -
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 -
Uniform m(t):
m(t) = ? -
Uniform m(t):
m(t) = (e^tΘ2 - e^tΘ1) /
t(Θ2 - Θ1) -
Exp m(t):
m(t) = ? -
Exp m(t):
m(t) = (1 - βt)^-1 -
Chi-Square m(t):
m(t) = ? -
Chi-Square m(t):
m(t) = (1 - 2t)^-ν/2 -
Taylor Series:
e^x = ? -
Taylor Series:
e^x = Σ x^i / i!
[i=0,∞] -
Binomial Expansion:
(x + y)^n = ? -
Binomial Expansion:
(x + y)^n = Σ(n C i)(x^n-i)(y^i)
[i=0,n] -
Series Expansion
ln(1+x) = ? -
Series Expansion
ln(1+x) = x - x^2/2 + x^3/3 - x^4/4 + ... -
Natural Log of Power
ln(x^2) = ? -
Natural Log of Power
ln(x^2) = 2ln(x) -
e^x = ?
n->∞ -
e^x = lim(1+x/n)^n
n->∞ -
Definition of e
e^x = ?
n->∞ -
Definition of e
e^x = lim(1-x/n)^x
n->∞