Lecture 10: Jan 28. Independent random variables Ross 6.2
Xand Yare independent if for any subsets Aand Bof ℜ,P(X∈A∩Y∈B) = P(X∈A)×P((Y∈B).
10.1 Independence of two r.vs: discrete case
Discrete case: this is equivalent to pX,Y (x, y) = P(X=x, Y =y) = P(X=x).P (Y=y) = pX(x)pY(y).
Clearly this is necessary: take A={x}and B={y}.
Conversely, if pX,Y (x, y) = pX(x)pY(y), then for any A,B:
P(X∈A, Y ∈B) = X
x∈AX
y∈B
pX,Y (x, y) = X
x∈AX
y∈B
pX(x)pY(y)
=X
x∈A
pX(x)X
y∈B
pY(y) = P(X∈A)P(Y∈B)
Note this must hold for all x, y. Thus the ranges of the r.v.s cannot depend on each other. Example: Xis
value on first die, Yon second: these are independent. But if we throw out doubles (i.e. points with x=y)
they are no longer independent: if X= 4 we know Y6= 4.
10.2 Independence of two r.vs: continuous case
With A= (−∞, x) and B= (−∞, y) we see FX,Y (x, y) = FX(x)FY(y).
As with the 1-dimensional case, this is also sufficient:
Xand Yare independent if and only if FX,Y (x, y) = FX(x)FY(y) for all x, y .
Differentiating, we see this means fX,Y (x, y) = fX(x)fY(y), and conversely integrating we see these are
equivalent. Note again it must hold for all x, y: the ranges of the r.vs cannot depend on each other.
10.3 Buffon’s needle: a classic example of estimating π.
Parallel lines distance Dapart; needle length L, with L≤D.
Let Xbe distance from needle midpoint to nearest line; θbe angle of needle to X.Xis U(0, D/2), θis
U(0, π/2) and they are independent. We want the probability
P(X≤(L/2) cos(θ)) = Z Z2x<L cos(y)
fX(x)fθ(y)dx dy =2
π
2
DZπ/2
0ZL/2 cos(y)
0
dx dy
=4
π D Zπ/2
0
L
2cos(y)dy =2L
π D[sin(y)]π /2
0= 2L/πD
10.4 Examples of independence and dependence
Xand Yare independent if and only if fX,Y =fX(x)fY(y) for all x, y.
Also, if fX,Y =g1(x)g2(y) for all X, Y , then Xand Yare independent, and fX(x)∝g1(x), fY(y)∝g2(y).
(i) Example: fX,Y (x, y ) = x+yon 0 < x < 1, 0 < y < 1. (Ch 6: # 22)
Xand Yare not independent: why not?
(ii) Example: fX,Y (x, y ) = exp(−(x+y)) on 0 < x < ∞, 0 < y < ∞.
Xand Yare independent: why? What is fX(x) ?
(iii) Example: fX,Y (x, y ) = 2 exp(−(4x+1
2y)) on 0 < x < ∞, 0 < y < ∞.
Xand Yare independent: why? What is fX(x) ?
(iv) Example: fX,Y (x, y ) = 1/6 on 0 < x < 2, 0 < y < 3,
Xand Yare independent: why? What is fX(x) ?
(v) Example: fX,Y (x, y ) = 2 on 0 < x, 0 < y,x+y < 1
Xand Yare not independent: why not?
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