• This module reviews statistical concepts, definitions, and theories from previous STAT courses, serving as a refresher for STAT 146.
• By the end of this unit, one should be able to define, compute, identify, describe, and explain various scenarios using random variables, expectations, and distributions.

Key Definitions

• Target Population: All entities or elements under study
• Sample: A subset of a target population
• Sampling: The process of selecting a target population sample
• Random Sample: A sample of size n (X1, ..., Xn) from a population with density function f(.), where samples are independent and have the same distribution
• Sampled Population: Population from which a random sample is obtained
• Statistic: Observable random variable function without unknown parameters
• Distribution Function: Probability that a random variable is less than or equal to x
• Random Variable: Rule assigning a numeric value to each experiment outcome
  • Discrete: countable values
  • Continuous: any value over an interval

PMF and PDF Functions

• Probability Mass Function (PMF): Px(xi) = P[X = x₁] ≥ 0, ∀xi; ΣPx(xi) = 1; used for discrete variables
• Probability Density Function (PDF): Used for continuous variables
  • fx(x) ≥ 0, ∀x; integral of fx(x)dx = 1

Distribution Function Properties

• Fx(x) = P[w: −∞ < X(w) ≤ x] = P(X ≤ x), ∀x ∈ R
• Properties: 0 ≤ Fx(x) ≤ 1, ∀x ∈ R; non-decreasing; right continuous; limit of Fx(x) as x approaches infinity is 1 and negative infinity is 0
• PMF/PDF relation to Distribution Function:
  • Discrete: Fx(x) = P[X ≤ x] = ∑xi≤xP[X = xi]; P[X = x] = F(x) – F(x¯)
  • Continuous: Fx(x) = integral from -inf to x of fx(x)dx; fx(x) = dF(x)/dx

Expected Values

• g(X) is a random variable,
  • If g(X) is discrete: E[g(X)] = Σ g(xi)P[X = xi]
  • If g(X) is continuous: E[g(X)] = integral from -inf to inf of g(X)f(X)dX

Useful Expectations of a Random Variable

• E[X], denoted as μ, represents the first moment (mean) of X
• E[X-C] represents the first moment about c
• E[(X-C)^k] represents the kth moment about c
• E[(X-μ)^k] is the kth central moment of X.
• E[(X-μ)^2] is variance (σ^2)
• E[(X-μ)^3] is skewness of X (μ3)
• E[(X-μ)^4] is kurtosis of X (μ4)
• E[X^K] is the kth (raw) moment of X (mk)
• E[ | X | ^ K] is the kth absolute moment of X

Expectation Properties

• E[c] = c, where c is a constant
• E[cg(X)] = cE[g(X)]
• E[c₁g(X) + c₂] = c₁E[g(X)] + C₂
• E[c1g1(X) + c2g2(X)] = c1E[g1(X)] ± c2E[g2(X)]
• E[g₁(X)] ≤ E[g2(X)] if g₁(X) ≤ g2(X)
• E[g1(X) * g2(X)] = E[g1(X)]E[g2(X)] if g1(X) and g2(X) are independent

Variance

• V(X) = σ² = E[(X – μ)²] = E[X2] – (E[X])2
• V[c] = 0 where c is a constant
• V[cX] = c2V(X)
• V[C₁X] + V[c2] = c}V[X]

Multivariate Expectations

• Multivariate Expectations with Joint Density Function fx,y(x, y)
  • E[X] = double integral of xfx,y(x,y)dydx
  • E[Y] = double integral of yfx,y(x, y)dxdy
  • E[g(X)] = double integral of (x) fx,y(x, y)dydx
  • E[h(Y)] = double integral of (y) fx,y(x, y)dxdy
  • E[g(X,Y)] = double integral of g(x, y) fx,y(x, y)dxdy
• Variables X and Y are independent if:
  • f XY(x, y) = fX(x)fY(y);
  • cov (X,Y) = 0;
  • E(XY) = E(X)E(Y);
• Independent random variables are uncorrelated and uncorrelated random variables don't have to be independent

Discrete Distributions

Discrete Uniform

• Equally likely values and values are uniformly distributed
• P[X = x] = 1/N, x = 1,2,...N
• E[X] = (N+1)/2
• V[X] = (N^2 -1)/12

Bernoulli

• P[X = x] = p^x(1-p)^(1-x), x = 0,1,
• E[X] = p
• V[X] = p(1-p)

Binomial

• Random variable represents the number of success/failures in n trials
• E[X] = np
• V[X] = np(1 – p)

Hypergeometric

• Distribution represents the number of success/failures in n draws OR number of success/failures in a sample of size n
• E[X] = nD/N
• V[X] = n(D/N)((N-D)/(N-1))*((N-n)/(N-1))

Geometric

• Represents the number of trials until the first success
• E[X] = 1/p
• V[X] = (1-p)/(p^2)

Negative Binomial

• Represents the number of trials before the first success
• E[X] = r/p
• V[X] = r(1-p)/(p^2)

Poisson

• of success/failure/events in an interval

• E[X] = lambda
• V[X] = lambda

Multivariate Hypergeometric

• (See document for equations)

Multinomial

• Defining the random variables is the same with the binomial distribution. But use this distribution if the experiment has more than 2 outcomes and the sampling is done with replacement

Continuous Distributions

Continuous Uniform

• E[X] = (a+b)/2
• V[X] = (b-a)^2 / 12
• Random variable is evenly distributed

Exponential

• Time until the first Poissonian event occurs
• Only parameter is the failure rate, lambda, which is constant
• E[X] = 1/lambda
• V[X] = 1/lambda^2

Gamma

• The length of time until the rth success
• (See document for equations)

Weibull

• Used in failure time data (analyzing potential failures) distribution but does not assume constant failure rate

Beta

• (See document for equations)

Normal

• Symmetric distribution used in statistics
• E[X] = mu
• V[X] = sigma^2

Chi-Square

• Has v=degrees of freedom

Functions

• Gamma
• Beta