If the random variable x is exponentially distributed with parameter λ = 1.5,then the probability P(2 ≤ X ≤ 4)is equal to:
A)0.6667
B)0.5000
C)0.0473
D)0.2500

Question

Quizplus · Accepted Answer

The probability density function of an exponential distribution with parameter λ is given by:

f(x) = λe^(-λx), for x ≥ 0

So, for λ = 1.5, we have:

f(x) = 1.5e^(-1.5x), for x ≥ 0

The cumulative distribution function is:

F(x) = ∫₀ˣ f(t) dt = 1 - e^(-λx)

So, for λ = 1.5, we have:

F(x) = 1 - e^(-1.5x)

Now, to calculate P(2 ≤ X ≤ 4), we need to subtract the cumulative probabilities at x=2 from the cumulative probability at x=4:

P(2 ≤ X ≤ 4) = F(4) - F(2)
             = (1 - e^(-6)) - (1 - e^(-3))
             = e^(-3) - e^(-6)
             ≈ 0.0473

Therefore, the answer is C.

If the Random Variable X Is Exponentially Distributed with Parameter