Level 225
Level 227

#### 52 words 0 ignored

Ready to learn
Ready to review

## Ignore words

Check the boxes below to ignore/unignore words, then click save at the bottom. Ignored words will never appear in any learning session.

**Ignore?**

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