3. General Random Variables Part II: Exponen9al Random Variable ECE 302 Spring 2012 Purdue University, School of ECE Prof. Ilya Pollak Exponen9al r.v. ⎧⎪ λ e− λ x , if x ≥ 0
f X (x) = ⎨
⎪⎩ 0, otherwise
Here, λ > 0.
fX(x)
λ
x
Ilya Pollak
Exponen9al r.v. ⎧⎪ λ e− λ x , if x ≥ 0
f X (x) = ⎨
⎪⎩ 0, otherwise
Here, λ > 0.
∞
Note that
∫
−∞
∞
f X (x)dx = ∫ λ e
−λx
dx = −e
−λx ∞
0
= 1.
0
Ilya Pollak
Mean of an exponen9al r.v. ⎧⎪ λ e− λ x , if x ≥ 0
f X (x) = ⎨
⎪⎩ 0, otherwise
∞
E[X] =
∫
∞
−∞
xf X (x)dx = ∫ xλ e− λ x dx
0
∞
∞
= ⎡⎣ −xe− λ x ⎤⎦ 0 + ∫ e− λ x dx
 0
(integration by parts)
0
−λx
∞
⎡e ⎤
1
=⎢
=
⎥
⎣ − λ ⎦0 λ
b
b
Recall: ∫ uv'dx = uv a − ∫ u 'v dx
b
a
a
Above, u = x, v = −e− λ x , so v' = λ e− λ x and u ' = 1
Ilya Pollak
Second moment of an exponen9al r.v. E ⎡⎣ X 2 ⎤⎦ =
∞
2
−λx
x
λ
e
dx
∫
0
[integrate by parts with u = x 2 , v = −e− λ x , v' = λ e− λ x and u ' = 2x]
∞
∞
= ⎡⎣ −x 2 e− λ x ⎤⎦ 0 + ∫ 2xe− λ x dx
0
∞
2
= 0 + ∫ xλ e− λ x dx
λ0

E[ X ]=
1
λ
2
= 2
λ
Ilya Pollak
Variance and standard devia9on of an
exponen9al r.v. 2
1
1
var(X) = E ⎡⎣ X ⎤⎦ − ( E[X]) = 2 − 2 = 2
λ
λ
λ
1
σ X = var(X) =
λ
2
2
Ilya Pollak
Geometric CDF •  Every δ seconds, flip a coin with P(H)=p. •  All flips are independent. •  Let X = 9me un9l first H. Then ⎧⎪ p(1 − p)k −1 , k=1,2,…
p X (kδ ) = ⎨
⎪⎩ 0, otherwise
n
n
k =1
k =1
FX (nδ ) = P(X ≤ nδ ) = ∑ p X (kδ ) = ∑ p(1 − p)k −1
n
1
−
(1
−
p)
= p ∑ (1 − p)m = p
= 1 − (1 − p)n , n=1,2,…
1 − (1 − p)
m=0
If x < δ , FX (x) = 0
If x ≥ δ , FX (x) = FX (nδ ) for nδ ≤ x < (n + 1)δ
n −1
Ilya Pollak
Geometric CDF ⎧⎪ 1 − (1 − p)n , for nδ ≤ x < (n + 1)δ , n=1,2,…
FX (x) = ⎨
⎪⎩ 0, for x < δ
FX(x)
1
δ 2δ 3δ 4δ
x
Ilya Pollak
Exponen9al CDF ⎧λ e- λ x , for x > 0
fY (x) = ⎨
⎩0, for x ≤ 0
⎧ 0, for x ≤ 0
⎪ x
FY (x) = ⎨
- λt
- λt x
-λ x
λ
e
dt
=
−
e
=
1
−
e
, for x > 0
⎪ ∫
0
⎪⎩ 0
FY(x)
1
x
Ilya Pollak
Exponen9al and Geometric CDFs ⎧⎪ 1 − (1 − p)n , for nδ ≤ x < (n + 1)δ , n=1,2,…
⎧1 − e- λ x , for x > 0
FY (x) = ⎨
FX (x) = ⎨
0,
for
x
≤
0
⎩
⎪⎩ 0, for x < δ
ln(1 − p)
Defining δ = −
, we have e- λδ = 1 − p, and therefore
λ
⎧⎪ 0, for n ≤ 0
⎧⎪ 0, for n ≤ 0
FY (nδ ) = ⎨
=⎨
= FX (nδ )
n
- λδ n
⎪⎩ 1 − e , for n ≥ 1
⎪⎩ 1 − (1 − p) , for n ≥ 1
1
FY(x)
Geometric CDF with p = 1 − e- λδ
δ 2δ 3δ 4δ
As δ → 0, FX → FY
x
ln(1 − p)
is the limit
δ
of the geometric, as p → 0 and # experiments per unit time → ∞.
In this sense, the exponential r.v. with parameter λ = −
Ilya Pollak