Exam 2 Study Guide

For the Exam on Wednesday, March 5.

Calculator Website Allowed for the Exam

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1. Discrete Random Variables (RV), Mass Function, CDF, Expected Value, and Variance

• Definition: A discrete random variable (RV) takes on a countable number of values.
• Probability Mass Function (PMF): P(X=x)P(X = x)
• Cumulative Distribution Function (CDF): F(x)=P(X≤x)F(x) = P(X \leq x)
• Expected Value: E[X]=∑xP(X=x)E[X] = \sum x P(X = x)
• Variance: Var(X)=E[X2]−(E[X])2Var(X) = E[X^2] - (E[X])
• Properties: Linearity of expectation, rules for variance calculation

2. Discrete RV’s Joint and Conditional Distributions

• Joint PMF: P(X=x,Y=y)P(X = x, Y = y)
• Marginal PMF: P(X=x)=∑yP(X=x,Y=y)P(X = x) = \sum_y P(X = x, Y = y)
• Conditional PMF: P(X=x∣Y=y)=P(X=x,Y=y)P(Y=y)P(X = x | Y = y) = \frac{P(X = x, Y = y)}{P(Y = y)}
• Independence: P(X=x,Y=y)=P(X=x)P(Y=y)P(X = x, Y = y) = P(X = x) P(Y = y)
• Expectation and Covariance: Cov(X,Y)=E[XY]−E[X]E[Y]Cov(X, Y) = E[XY] - E[X]E[Y]

3. Bernoulli Distribution

• Definition: A discrete RV that takes values 1 (success) with probability pp and 0 (failure) with probability 1−p1 - p.
• PMF: P(X=x)=px(1−p)1−x,x∈{0,1}P(X = x) = p^x (1-p)^{1-x}, x \in {0,1}
• Expectation: E[X]=pE[X] = p
• Variance: Var(X)=p(1−p)Var(X) = p(1 - p)

4. Binomial Distribution

• Definition: The sum of nn independent Bernoulli trials, each with success probability pp.
• PMF: P(X=k)=(nk)pk(1−p)n−kP(X = k) = \binom{n}{k} p^k (1-p)^{n-k}, for k=0,1,…,nk = 0,1,…,n
• Expectation: E[X]=npE[X] = np
• Variance: Var(X)=np(1−p)Var(X) = np(1 - p)

5. Geometric Distribution

• Definition: The number of trials until the first success in a sequence of independent Bernoulli trials.
• PMF: P(X=k)=(1−p)k−1pP(X = k) = (1-p)^{k-1} p, for k=1,2,…k = 1,2,…
• Expectation: E[X]=1pE[X] = \frac{1}{p}
• Variance: Var(X)=1−pp2Var(X) = \frac{1 - p}{p^2}

6. Negative Binomial Distribution

• Definition: The number of trials until the rrth success in a sequence of independent Bernoulli trials.
• PMF: P(X=k)=(k−1r−1)pr(1−p)k−rP(X = k) = \binom{k-1}{r-1} p^r (1-p)^{k-r}, for k=r,r+1,…k = r, r+1,…
• Expectation: E[X]=rpE[X] = \frac{r}{p}
• Variance: Var(X)=r(1−p)p2Var(X) = \frac{r(1 - p)}{p^2}

7. Poisson Distribution

• Definition: Models the number of events occurring in a fixed interval, given the rate λ\lambda.
• PMF: P(X=k)=e−λλkk!P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}, for k=0,1,2,…k = 0,1,2,…
• Expectation: E[X]=λE[X] = \lambda
• Variance: Var(X)=λVar(X) = \lambda
• Poisson Approximation: Approximates Binomial when nn is large and pp is small with λ=np\lambda = np.

Study Tips

• Understand definitions and derive key formulas.
• Solve practice problems for each distribution.
• Work with joint and conditional probabilities.
• Familiarize yourself with expectation and variance calculations.
• Use properties of independence and covariance to simplify problems.

stateDiagram
    [*] --> NULL: Start
    NULL --> ID: SELECT let go
    ID --> NULL: SELECT held for one second
    NULL --> YSM: Accounts exist
    NULL --> [*]: No accounts stored

    YSM --> PEN: RIGHT Pressed
    PEN --> MSM: RIGHT Pressed
    MSM --> TAX: RIGHT Pressed
    TAX --> MPC: RIGHT Pressed

    PEN --> YSM: LEFT Pressed
    MSM --> PEN: LEFT Pressed
    TAX --> MSM: LEFT Pressed
    MPC --> TAX: LEFT Pressed