10.1.1 PDFs and CDFs
Consider the random process
{X(t),t∈J}. For any
t0∈J,
X(t0) is a random variable, so we can write its CDF
FX(t0)(x)=P(X(t0)≤x).
If
t1,t2∈J, then we can find the joint CDF of
X(t1) and
X(t2) by
FX(t1)X(t2)(x1,x2)=P(X(t1)≤x1,X(t2)≤x2).
More generally for
t1,t2,⋯,tn∈J, we can write
FX(t1)X(t2)⋯X(tn)(x1,x2,⋯,xn)=P(X(t1)≤x1,X(t2)≤x2,⋯,X(tn)≤xn).
Similarly, we can write joint PDFs or PMFs depending on whether
X(t) is continuous-valued (the
X(ti)'s are continuous random variables) or discrete-valued (the
X(ti)'s are discrete random variables).
Example
Consider the random process
{Xn,n=0,1,2,⋯}, in which
Xi's are i.i.d. standard normal random variables.
- Write down fXn(x) for n=0,1,2,⋯.
- Write down fXmXn(x1,x2) for m≠n.
- Solution
-
- Since Xn∼N(0,1), we have
fXn(x)=12π−−√e−x22, for all x∈R.
- If m≠n, then Xm and Xn are independent (because of the i.i.d. assumption), so
fXmXn(x1,x2)=fXm(x1)fXn(x2)=12π−−√e−x212⋅12π−−√e−x222=12πexp{−x21+x222}, for all x1,x2∈R.
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Practical uncertainty: Useful Ideas in Decision-Making, Risk, Randomness, & AI
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