Let X be an exponential random variable with parameter lambd

Let X be an exponential random variable with parameter lambda(just use \"a\" to represent it).

(a) For d >0 and k a nonnegative integer. Find P[kd

(b) Segment the positive real line into four equiprobable disjoint intervals.

Solution

As here parameter =2

f(x) = 2e^-2x, x>-0, 0 otherwise.

P(Kd

= -e^-2x

Substitute limits

Reqd prob = e^-2d(k+1)-e^-2kd

^{-----------------------------------------------------------------------------------------}

^{Consider P(xc) =0.25}

^{Then a, b, c, partition the intervals into 4 equiprobable parts}

^{P(xf(x) dx from 0 to a = -e-2c+1 = 0.25}

or 0.75 = e^-2a

a=0.1438

To find b:

Using the same process as above e^-2a-e^{-2b = 0.75-}e^{-2b = 0.25}

^{b = 0.3466}

^{To find c:}

Using the same process as above e^-2b-e^{-2c =} 0.50^-e^{-2c = 0.25}

c = 0.6931