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Level 226

Probability III


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Poisson Distribution
Used to find the probability that an event happens in a given interval.
Moment About the Origin
E(Xⁿ) = The nth moment of X about the origin.
Moment About the Mean
E[(X - µ)ⁿ] = The nth moment of X about its mean.
Moment Generating Function of a DRV
Mgf(t) = E(e^tX), -δ < t < δ,
Tchebysheff's Theorem
Used to determine the lower bound for the probability that a random variable Y of interest falls in an interval µ ± kσ.
Continuous Random Variables
A variable that may take on an infinite number of potential outcomes.
Distribution Function
F(x) = P(X ≤ x), x ∈ R
Properties of Distribution Functions
- The distribution function F is non-decreasing
P[a ≤ X ≤ b]
P[a ≤ X ≤ b] = F(b) - F(a) = ∫(a to b) f(x)dx
Density Function
f(t), where distribution function F(x) = ∫(-∞ to x) f(t)dt
Properties of Density Functions
- f(x) ≥ 0 for all x ∈ R
Expected Value of a CRV
E(X) = ∫(-∞ to ∞) x * f(x) dx
Variance of a CRV
E[(x - µ)²] = ∫(-∞ to ∞) (x - µ)² dx
Var(aX + b) - CRV
Var(aX + b) = a²Var(X)
Moment Generating Function of a CRV
Mgf(t) = E(e^tx) = ∫(-∞ to ∞) e^tx * f(x) dx
Uniform Distribution
f(x) =
Properties of the Uniform Distribution
E(x) = b+a / 2 (midpoint)
Exponential Distribution
f(x) =
E(x) = β
Properties of the Exponential Distribution
Gamma Function
Γ(α) = ∫(0 to ∞) x^(α-1) * e^-x, where α > 0
- Γ(1) = 1
Properties of the Gamma Function
Gamma Distribution
f(x) =
E(X) = αβ
Properties of the Gamma Distribution
Normal (Gaussian) Distribution
f(x) = (1 / σ√2π) * e(-1/2 (x-µ / σ)²), -∞ < x < ∞
Z-score
z is the distance from the mean of the normal distribution expressed in units of standard deviation.
E(X) = µ
Properties of the Normal Distribution
Beta Distribution
f(x)
Properties of the Beta Distribution
E(Y) = µ = α / α+β
Markov's Inequality
P[X > ε] ≤ E(X) / ε, for any ε > 0
Chebychev's Inequality
P[|X - µ| > ε] = σ² / ε², for any ε > 0
Bivariate Joint Probability Function
f(y₁, y₂) = P[Y₁ = y₁, Y₂ = y₂], -∞ < y₁, y₂ < ∞
Properties of DRVs with Joint Probability Functions f(y₁, y2)
f(y₁, y₂) ≥ 0 for all y₁, y₂ ∈ R
Bivariate Joint Distribution Function
F(y₁, y₂) = P(Y₁ ≤ y₁, Y₂ ≤ y₂), -∞ < y₁, y₂ < ∞
Properties of Bivariate Joint Distribution Functions
F(-∞, -∞) = F(y₁, -∞) = F(-∞, y₂) = 0
Bivariate Joint Probability Density Function
If there exists a function,
Properties of Bivariate Joint Probability Density Functions
f(y₁, y₂) ≥ 0 for all y₁, y₂
Finding the probability given a Bivariate Joint Probability Density Function
P[a₁ < Y₁ ≤ b₁, a₂ < Y₂ ≤ b₂] = ∫(b₁ to b₂)∫(a₁ to a₂) f(y₁, y₂) dy₁ dy₂
Marginal Probability Functions
p₁(y₁) = ∑(all y₂) p(y₁, y₂)
Marginal Density Functions
f₁(y₁) = ∫(-∞ to ∞) f(y₁, y₂) dy2
Bivariate Conditional Discrete Probability Function
p(y₁|y₂) = P(Y₁ = y₁|Y₂ = y₂)
Bivariate Conditional Distribution Function
F(y₁|y₂) = P(y₁ ≤ y₁|Y₂ = y₂)
Properties of Bivariate Independent Random Variables
Y₁ and Y₂ are independent iff,
Expected Value of a Function of Two Random Variables
If g(Y₁, Y₂,..., Yk) is a function of DRVs with a probability function p(y₁, y₂, ..., yk),
E(c) = c
Special Theorems for the Expected Value of Two Random Variables
Covariance of Two Random Variables
Cov(Y₁, Y₂) = E[Y₁ - µ₁)(Y₂ - µ₂)] = E(Y₁Y₂) - E(Y1)E(Y₂) = E(Y₁Y₂) - µ1µ2
Cov(Y₁, Y2) = 0
Covariance of Independent Random Variables
Linear Function of Random Variables
U1 = a₁Y₁ + a₂Y₂ + ... + anYn = ∑(i=1 to n) aiYi,
Properties of Linear Functions of Random Variables
E(U1) = ∑(i=1 to n) a₋i * µ₋i
correlation coefficient
If σ₁ and σ₂ are the variances of Y₁ and Y₂, the correlation coefficient is:
Multinomial Experiment
An experiment with n independent trials, where each trial may produce one of k outcomes. We are interested in the random variables Y₁, Y₂, ..., Yk, where Yi is the number of trials that produce outcome i.
Multinomial Joint Probability Function
p(y₁, y₂, ..., yk) = P[Y₁ = y₁, ..., Y₋k = yk] = (n! / y₁!y₂!...yk!) * p₁^(y₁) *...*p₋k^(y₋k),
Chi-Square Distribution
A gamma distributed random variable with α = v/2, and β = 2