Probability Distribution Reference Sheet  
MATH 394: Probability I (Summer 2026)  
Instructor: Arman Jahangiri, Department of Mathematics @ University of Washington  
Discrete Distributions  
Name  
Parameters  
Probability Mass Function  
Support  
Mean  
Variance  
MGF  
Bernoulli  
0 ≤ p ≤ 1  
P(X = k) = pk(1 − p)1−k  
x = 0, 1  
p
p(1 − p)  
1 − p + pet  
ꢀ ꢁ  
n
k
Binomial  
Poisson  
n ≥ 1, 0 ≤ p ≤ 1 P(X = k) =  
pk(1 − p)n−k  
k = 0, . . . , n  
k = 0, 1, 2, . . .  
k = 1, 2, . . .  
np  
λ
np(1 − p)  
(1 − p + pet)n  
λk  
λ > 0  
P(X = k) = e−λ  
λ
exp{λ(et − 1)}  
k!  
1
1 − p  
pet  
Geometric  
0 < p ≤ 1  
P(X = k) = p(1 − p)k−1  
p
p2  
1 − (1 − p)et  
1 − p  
1 − p  
p
Geometric,  
zero-based  
0 < p ≤ 1  
P(X = k) = p(1 − p)k  
k = 0, 1, 2, . . .  
p
p2  
1 − (1 − p)et  
ꢂ
ꢃ
ꢃ
r
r
pet  
ꢀ
ꢀ
ꢁ
r
r(1 − p)  
k−1  
Negative  
Binomial  
r ≥ 1, 0 < p ≤ 1 P(X = k) =  
r ≥ 1, 0 < p ≤ 1 P(X = k) =  
pr(1 − p)k−r k = r, r + 1, . . .  
r−1  
p
p2  
1 − (1 − p)et  
ꢂ
ꢁ
r(1 − p)  
r(1 − p)  
p
r+k−1  
Negative  
Binomial,  
failures before  
r successes  
pr(1 − p)k k = 0, 1, 2, . . .  
r−1  
p
p2  
1 − (1 − p)et  
ꢀ ꢁꢀ  
ꢁ
ꢀ ꢁꢀ  
ꢁ
ꢂ
ꢃ
w
k
b
w
k
b
X
nw  
w + b − n  
w
w
n−k  
n−k  
Hypergeometric w, b, n  
P(X = k) =  
ꢀ
ꢁ
max(0, n − b) ≤  
n
1 −  
etk  
ꢀ
ꢁ
w+b  
n
w+b  
n
w + b  
w + b − 1 w + b  
w + b  
k ≤ min(n, w)  
k
ꢀ
ꢁꢀ  
ꢁ
b
r+k−1 w+b−r−k  
X
rb  
rb(w + b + 1)(w − r + 1)  
r−1  
w−r  
Negative Hy-  
pergeometric  
w, b, r  
P(X = k) =  
k = 0, 1, . . . , b  
etk  
ꢀ
ꢁ
w+b  
w
(w + 1)2(w + 2)  
ꢀ
ꢁꢀ  
ꢁ
w + 1  
r+k−1 w+b−r−k  
k=0  
r−1  
w−r  
ꢀ
ꢁ
w+b  
w
© 2026 Arman Jahangiri — Department of Mathematics @ University of Washington.  
For educational use in MATH 394: Probability I  
Continuous Distributions  
Name  
Parameters  
Probability Density Function  
Support  
Mean  
Variance  
MGF  
1
fX (x) =  
b − a  
a + b  
(b − a)2  
etb − eta  
Uniform  
a < b  
a < x < b  
2
12  
t(b − a)  
ꢄ
ꢅ
σ2t2  
Normal  
µ ∈ R, σ2 > 0  
fX (x) =  
−∞ < x < ∞  
µ
σ2  
exp µt +  
ꢄ
ꢄ
ꢅ
2
1
(x − µ)2  
√
exp  
−
2σ2  
2πσ2  
2
2
2
Log-Normal  
µ ∈ R, σ2 > 0  
fX (x) =  
x > 0  
eµ+σ /2  
e2µ+σ (eσ − 1)  
DNE  
ꢅ
1
(log x − µ)2  
√
exp  
−
2σ2  
xσ 2π  
1
1
λ2  
λ
Exponential  
Gamma  
Beta  
λ > 0  
fX (x) = λe−λx  
x ≥ 0  
x ≥ 0  
λ
λ − t  
ꢂ
ꢃ
r
λr  
r
r
λ2  
λ
r > 0, λ > 0  
a, b > 0  
fX (x) =  
xr−1e−λx  
Γ(r)  
λ
λ − t  
Γ(a + b)  
a
ab  
fX (x) =  
xa−1(1 − x)b−1  
0 < x < 1  
x > 0  
–
Γ(a)Γ(b)  
a + b  
(a + b)2(a + b + 1)  
α
Γ(1 + 1/α)  
Γ(1 + 2/α)  
Weibull  
α, λ > 0  
fX (x) = αλxα−1e−λx  
−
–
λ1/α  
λ2/α  
µ2  
1
Chi-Square  
n > 0  
fX (x) =  
xn/2−1e−x/2  
x > 0  
n
2n  
(1 − 2t)−n/2  
2n/2Γ(n/2)  
n
Student-t  
n > 0  
fX (x) =  
−∞ < x < ∞  
0, n > 1  
, n > 2  
DNE  
n − 2  
ꢂ
ꢃ−(n+1)/2  
Γ((n + 1)/2)  
√
x2  
n
1 +  
nπΓ(n/2)  
Moment-Generating Function  
The moment-generating function is  
MX(t) = E(etX).  
When it exists near t = 0, it can be used to compute moments:  
E(Xn) = MX(n)(0),  
© 2026 Arman Jahangiri — Department of Mathematics @ University of Washington.  
For educational use in MATH 394: Probability I