University of Washington
Department of Mathematics
MATH 394: Probability I
Probability Distribution Reference Sheet
MATH 394: Probability I (Summer 2026)
Instructor: Arman Jahangiri, Department of Mathematics @ University of Washington
Discrete Distributions I
Quantity
Bernoulli
Binomial
Poisson
Geometric
Geometric,
zero-based
Parameters
PMF
0 ≤ p ≤ 1
n ≥ 1, 0 ≤ p ≤ 1
λ > 0
0 < p ≤ 1
0 < p ≤ 1
λk
P(X = k) =
pk(1 − p)1−k
k = 0, 1
P(X = k) =
pk(1 − p)n−k
k = 0, . . . , n
P(X = k) = e−λ
P(X = k) = p(1−p)k−1 P(X = k) = p(1 − p)k
ꢀ ꢁ
n
k
k!
Support
Mean
k = 0, 1, 2, . . .
k = 1, 2, . . .
1
k = 0, 1, 2, . . .
1 − p
p
np
λ
p
p
1 − p
1 − p
Variance
MGF
p(1 − p)
np(1 − p)
λ
p2
p2
p
pet
1 − p + pet
(1 − p + pet)n
exp{λ(et − 1)}
1 − (1 − p)et
1 − (1 − p)et
Discrete Distributions II
Quantity
Negative Binomial
Negative Binomial,
failures before r successes
Hypergeometric
Negative Hypergeometric
Parameters
PMF
r ≥ 1, 0 < p ≤ 1
r ≥ 1, 0 < p ≤ 1
w, b, n
w, b, r
ꢀ ꢁꢀ
ꢁ
w
b
k
n−k
P(X = k) =
pr(1 − p)k−r
P(X = k) =
P(X = k) =
P(X = k) =
r+k−1 w+b−r−k
ꢀ
ꢁ
w+b
n
ꢀ
ꢁ
ꢀ
ꢁ
ꢀ
ꢁꢀ
ꢁ
k−1
r+k−1
pr(1 − p)k
r−1
r−1
r−1
w−r
ꢀ
ꢁ
w+b
w
Support
Mean
k = r, r + 1, . . .
k = 0, 1, 2, . . .
max(0, n − b) ≤ k ≤ min(n, w)
k = 0, 1, . . . , b
r
r(1 − p)
nw
rb
p
p
w + b
w
w + 1
rb(w + b + 1)(w − r + 1)
ꢂ
ꢃ
r(1 − p)
r(1 − p)
w + b − n
w
Variance
MGF
n
1 −
p2
pet
p2
w + b − 1 w + b
w + b
(w + 1)2(w + 2)
ꢀ ꢁꢀ
ꢁ
ꢀ
ꢁꢀ
ꢁ
ꢂ
ꢃ
ꢂ
ꢃ
w
k
ꢀ
b
r+k−1 w+b−r−k
b
r
r
X
X
p
n−k
r−1
w−r
etk
etk
ꢁ
ꢀ
ꢁ
w+b
n
w+b
w
1 − (1 − p)et
1 − (1 − p)et
k
k=0
© 2026 Arman Jahangiri — Department of Mathematics @ University of Washington.
For educational use in MATH 394: Probability I
Summer 2026
Arman Jahangiri
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