Question

If F(x) is a cumulative distribution function of a continuous random variable x with p.d.f f(x) then F′(x) = __________

Answer 1

Concept :- The cumulative distribution function of a continuous random variable X can be expressed as integral of its probability density function ( P.d.f) fₓ as follows $\bold{F_X(x)=\int\limits_{-\infty}^{x}{f_X(t)}\,dt}}$

Here we have to find out F'(x) .
Means, F'(x) = dF(x)/dx
Differentiate above expression with respect to x
F'(x) = f(x)dx/dx - f(-∞)d(-∞)/dx
F'(x) = f(x) - 0 = f(x)

Hence, F'(x) = f(x)