Download Chapter 3 homework (chapter3-homework.docx)
Download Chapter 3 homework solution (chapter3-homework-solution_1.html)
Download Chapter 3 homework solution (chapter3-homework-solution_2.html)
PMF f(x)=P(X=x) gives the probability at each discrete point. CDF F(x)=P(X≤x) accumulates probabilities up to x. For discrete X, F(x) jumps at the support points and is flat between them.
Two things: (1) all f(x) ≥ 0, and (2) the total sums to 1. In Excel, if x are in A2:A10 and probabilities in B2:B10, check =SUM(B2:B10) equals 1 (within rounding).
Mean: E[X] = Σ x f(x). Variance: Var(X) = E[X^2] − (E[X])^2 where E[X^2] = Σ x^2 f(x). In Excel: =SUMPRODUCT(A2:A10,B2:B10) for E[X], and =SUMPRODUCT((A2:A10)^2,B2:B10) - (SUMPRODUCT(A2:A10,B2:B10))^2 for Var(X).
E[Y] = aE[X] + b and Var(Y) = a^2 Var(X). Shifts (b) don’t change spread; scaling by a stretches the spread by |a| (variance scales by a^2).
Use Binomial(n,p) when you count successes in n independent, identical trials with success probability p. Mean: np, Variance: np(1−p). Example: correct T/F answers when guessing.
Number of trials to the first success (support 1,2,3,…). PMF: P(X=k)=(1−p)^{k−1}p. Mean: 1/p. It’s memoryless: P(X>s+t | X>s)=P(X>t).
Negative Binomial counts trials to achieve r successes (geometric is the case r=1). PMF: P(X=k)=C(k−1,r−1)p^r(1−p)^{k−r}, mean: r/p, variance: r(1−p)/p^2. Binomial counts number of successes in a fixed number of trials.
Counts of events in a fixed interval with constant average rate λ and independent occurrences. PMF: e^{−λ}λ^x/x!, mean = variance = λ. Also a good approx. to Binomial when n is large and p is small with λ=np.
If X takes integers from a to b equally likely, then f(x)=1/(b−a+1), E[X]=(a+b)/2, and Var(X)={ (b−a+1)^2 − 1 }/12.
np ≥ 10 and n(1−p) ≥ 10 (use continuity correction).n large, p small, with λ=np (moderate λ).λ with mean = variance = λ.