Stat 110 - Cheat Sheet Page 2
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1
2
3
4Let X be the number of “successful” units drown out of
Proof.
n draws without replacement from a population of w + b
1
P (X
x) = P (F
(U )
x)
units, where w is the number of “successful” units in then
= P (Y
F (x)) = F (x)
X
HGeom(w, b, n).
w
Mean: E[X] = n
w+b
Theorem 2. Let X be a r.v. with CDF F
(x) then Y =
Poisson Distribution
X
F (X)
U nif [0, 1].
X
P ois(λ) and has PMF
Proof. It is clear that Y takes values in (0, 1), then P (Y
k
λ
y) = 0 for y
0 and P (Y
1) = 1 for y
1. So for y
(0, 1)
λ
P (X = k) =
e
k!
P (Y
y) = P (F (X)
y)
1
1
Let X be the count of a particular rare event for a set
= P (X
F
(y)) = F (F
(y)) = y
period of time, say an hour, which has an expectation of
So Y has a U nif (0, 1) CDF.
λ then X
P ois(λ).
Exponential Distribution
Mean: E[X] = λ.
Let X
Expo(λ), then X has pdf
Negative Binomial
λx
f
(x) = λe
,
x > 0
X
X
N Bin(r, p) and has PMF
and CDF
n + r
1
λx
F
(x) = 1
e
.
n
r
X
P (X = n) =
(1
p)
p
r
1
Relationship with the uniform: X =
log U
Expo(1) if
U
U nif [0, 1]. Key property: Memoryless: If you have
Let X be the number of failures before the first r suc-
waited t minutes in line gives you no idea of how much longer
cess when the probability of success is p, then X
you have to wait!!
N Bin(r, p).
Normal (Gaussian) Distribution
2
Let X
N (µ, σ
), then X has a pdf (the cdf is nasty)
r(1 p)
Mean: E[X] =
.
p
1
1
2
(x µ)
Fundamental Bridge Let I
be the indication of the occur-
f
(x) =
e
2 2
A
X
2
2πσ
rence of event A. Then the fundamental bridge is
Relationship to uniform: Complicated, so Box-Muller trans-
P (A) = E[I
].
form.
A
Key property: The most magical distribution, just pops up
Maths
2
x
everywhere! Also tells you that
e
dx =
2π, why?
1
Moment Generating Function (MGF)
n
a
=
for a < 1
1
a
n=0
Let X be any random variable, then the MGF of X is
n
x
x
=e
n!
tX
M
(t) = E[e
]
n=0
X
n
2
2
n
t
X
n j
j
n
x
y
=(x + y)
.
= E 1 + tX +
+ . . .
j
2!
j=0
2
2
t
E[X
]
= 1 + tE[X] +
+ . . .
Uniform Distribution
2!
Let U
U nif [a, b], then U has a pdf
Notice that the MGF may not be finite for all values of
1
t, but for it to “exist” we require it to be finite in a small
f
(u) =
U
neighborhood around t = 0.
b
a
The MGF really does give you all of the moments of X!
and CDF (if x
[a, b])
th
For example, if we wish to find the n
moment of X we
u
1
u
a
th
just take the n
derivative of the MGF with respect to
F
U (u) =
du =
.
=
b
1
b
1
t and evaluate it at t = 0
a
n
∂
Key property: Universality of the Uniform: Let U be any
M
(t)
= M
(0)
X
X
n
∂t
random variable with CDF F
(u) then F
(U )
U nif [0, 1].
U
U
t=0
n
n(n
1) . . . 1E[X
]
=
+ terms containing t
n!
1
Theorem 1. U
U nif [0, 1] and let X = F
(U ) then X
n
= E[X
]
has CDF F.
2
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