Let X be a continuous random variable with the following PDF

Let X be a continuous random variable with the following PDF

fX(x)={ce^4x      x0

          0            otherwise

where c is a positive constant

Find c.

Find the CDF of X, FX(x).

Find P(2<X<5).

Find EX.

Solution

a)

By normality,

Integral [c e^-4x dx]|(0, oo) = 1

(-c/4) e^-4x|(0, oo) = 0 - (-c/4) = 1

Thus,

c = 4 [answer]

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b)

F(x) = Integral [4 e^-4x dx]

F(x) = -e^-4x + C

As F(oo) = 1,

0 + C = 1

C = 1

Thus,

F(x) = 1 - e^-4x [ANSWER]

**************

c)

P(2<x<5) = F(5) - F(2) = (1-e^(-4*5)) - (1-e^(-4*2))

= 0.000335461 [ANSWER]

***************

D)

For exponential distributions, E(x) = 1/c. Thus,

E(x) = 1/4 or 0.25 [answer]