10.1.1 PDFs and CDFs

Consider the random process {X(t),tJ}. For any t0J, X(t0) is a random variable, so we can write its CDF
FX(t0)(x)=P(X(t0)x).
If t1,t2J, 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,,tnJ, 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.
  1. Write down fXn(x) for n=0,1,2,.
  2. Write down fXmXn(x1,x2) for mn.
  • Solution
      1. Since XnN(0,1), we have
        fXn(x)=12πex22, for all xR.
      2. If mn, then Xm and Xn are independent (because of the i.i.d. assumption), so
        fXmXn(x1,x2)=fXm(x1)fXn(x2)=12πex12212πex222=12πexp{x12+x222}, for all x1,x2R.




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