Statistics Assignment Help With Random variable and Distribution Function

Let X is a random variable on (S, B, P). Then the function:

FX(x) = P(X≤ x) = P{ ω :X(ω)≤x}, -∞<x<∞

Is called the distribution function (d.f.) of X.

If clarity permits, we may write F(x) instead of FX(x).

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3.2.1 Properties of Distribution function:

The properties which are common to all distribution functions are

Property 1:

If F is the d.f. of the r.v. X and if a< b, then

P(a<X≤ b) = F(b)-F(a)

Property 2:

If F is the d.f. of one- dimensional r.v. X, then

  • (i) 0≤F(x) ≤1,
  • (ii) F(x) ≤F(y)
  • (iii) If x<y

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