Mark’s Formula Sheet for Exam P
Discrete distributions
•Uniform, U(m)
–PMF: f(x) = 1
m, for x= 1,2, . . . , m
–µ=m+ 1
2and σ2=m2−1
12
•Hypergeometric
–PMF: f(x) = N1
xN2
n−x
N
n
–xis the number of items from the sample of nitems that are from group/type 1.
–µ=n(N1
N) and σ2=n(N1
N)(N2
N)(N−n
N−1)
•Binomial, b(n, p)
–PMF: f(x) = n
xpx(1 −p)n−x, for x= 0,1, . . . , n
–xis the number of successes in ntrials.
–µ=np and σ2=np(1 −p) = npq
–MGF: M(t) = [(1 −p) + pet]n= (q+pet)n
•Negative Binomial, nb(r, p)
–PMF: f(x) = x−1
r−1pr(1 −p)x−r, for x=r, r + 1, r + 2, . . .
–xis the number of trials necessary to see rsuccesses.
–µ=r(1
p) = r
pand σ2=r(1 −p)
p2=rq
p2
–MGF: M(t) = (pet)r
[1 −(1 −p)et]r=pet
1−qetr
•Geometric, geo(p)
–PMF: f(x) = (1 −p)x−1p, for x= 1,2, . . .
–xis the number of trials necessary to see 1 success.
–CDF: P(X≤k)=1−(1 −p)k= 1 −qkand P(X > k) = (1 −p)k=qk
–µ=1
pand σ2=1−p
p2=q
p2
–MGF: M(t) = pet
1−(1 −p)et=pet
1−qet
–Distribution is said to be “memoryless”, because P(X > k +j|X > k) = P(X > j).
