The Erik Jonsson School of Engineering and Computer Science

Chapter 4pp. 153-210

William J. Pervin

The University of Texas at Dallas

Richardson, Texas 75083

The Erik Jonsson School of Engineering and Computer Science

Chapter 3

Pairs of Random Variables

The Erik Jonsson School of Engineering and Computer Science

Chapter 4

4.1 Joint CDF:

The joint CDF FX,Y of RVs X and Y is

FX,Y(x,y) = P[X ≤ x, Y ≤ y]

The Erik Jonsson School of Engineering and Computer Science

Chapter 4

0 ≤ FX,Y(x,y) ≤ 1

If x1 ≤ x2 and y1 ≤ y2

then FX,Y(x1,y1) ≤ FX,Y(x2,y2)

FX,Y(∞,∞) = 1

The Erik Jonsson School of Engineering and Computer Science

Chapter 4

4.2 Joint PMF:

The joint PMF of discrete RVs X and Y is

PX,Y(x,y) = P[X = x, Y = y]

SX,Y = SX x SY

The Erik Jonsson School of Engineering and Computer Science

Chapter 4

For discrete RVs X and Y and any

B X x Y, the probability of the

event {(X,Y) B} is

P[B] = Σ(x,y)B PX,Y(x,y)

The Erik Jonsson School of Engineering and Computer Science

Chapter 4

4.3 Marginal PMF:

For discrete RVs X and Y with joint PMF PX,Y(x,y),

PX(x) = ΣySY PX,Y(x,y)

PY(y) = ΣxSX PX,Y(x,y)

The Erik Jonsson School of Engineering and Computer Science

Chapter 4

4.4 Joint PDF:

The joint PDF of continuous RVs X and Y is function fX,Y such that

FX,Y(x,y) = ∫–∞

y ∫–∞

x fX,Y(u,v)dudv

fX,Y(x,y) = ∂2FX,Y(x,y)/∂x∂y

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Chapter 4

fX,Y(x,y) ≥ 0 for all (x,y)

FX,Y(x,y)(∞,∞) = 1

The Erik Jonsson School of Engineering and Computer Science

Chapter 4

4.5 Marginal PDF:

If X and Y are RVs with joint PDF fX,Y, the marginal PDFs are

fX(x) = Int{fX,Y(x,y)dy,-∞,-∞}

fy(x) = Int{fX,Y(x,y)dx,-∞,-∞}

The Erik Jonsson School of Engineering and Computer Science

Chapter 4

4.6 Functions of Two RVs:

Derived RV W=g(X,Y)

X,Y discrete:

PW(w) = Sum{PX,Y(x,y)|(x,y):g(x,y)=w}

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Chapter 4

X,Y continuous:

FW(w) = P[W ≤ w] = ∫∫g(x,y)=w fX,Y(x,y)dxdy

Example: If W = max(X,Y), then

FW(w) = FX,Y(w,w) = ∫y≤w ∫x ≤w fX,Y (x,y)dxdy

The Erik Jonsson School of Engineering and Computer Science

Chapter 4

4.7 Expected Values:

For RVs X and Y, if W = g(X,Y) then

Discrete: E[W] = Σ Σ g(x,y)PX,Y(x,y)

Continuous: E[W] = ∫ ∫ g(x,y)fX,Y(x,y)dxdy

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Chapter 4

Theorem: E[Σgi(X,Y)] = ΣE[gi(X,Y)]

In particular: E[X + Y] = E[X] + E[Y]

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Chapter 4

The covariance of two RVs X and Y is

Cov[X,Y] = σXY = E[(X – μX)(Y – μY)]

Var[X + Y] = Var[X] + Var[Y] + 2Cov[X,Y]

The Erik Jonsson School of Engineering and Computer Science

Chapter 4

The correlation of two RVs X and Y is

rX,Y = E[XY]

Cov[X,Y] = rX,Y – μX μY

Cov[X,X] = Var[X] and rX,X = E[X2]

Correlation coefficient ρX,Y=Cov[X,Y]/σXσY

The Erik Jonsson School of Engineering and Computer Science

Chapter 4

4.10 Independent RVs:

Discrete: PX,Y(x,y) = PX(x)PY(y)

Continuous: fX,Y(x,y) = fX(x)fY(y)

The Erik Jonsson School of Engineering and Computer Science

Chapter 4

For independent RVs X and Y:

E[g(X)h(Y)] = E[g(X)]E[h(Y)]

rX,Y = E[XY] = E[X]E[Y]

Cov[X,Y] = σX,Y = 0

Var[X + Y] = Var[X] + Var[Y]