Probability & Statistics Ch.2 Section 4 Question 6

Solutions for “Probability and Statistics: The Science of Uncertainty” (Second Edition). These are solutions I have come up with; I offer no guarantee of accuracy.

Question

Let $X \sim \text{Exponential}(3)$ . Compute each of the following:

a) $P(0 < X < 1)$
b) $P(0 < X < 3)$
c) $P(0 < X < 5)$
d) $P(2 < X < 5)$
e) $P(2 < X < 10)$
f) $P(X > 2)$

Solution

From Example 2.4.5 in the textbook:

P(a \leq X \leq b) = \int_a^b \lambda e^{-\lambda x} \, dx = e^{-\lambda a} - e^{-\lambda b}

In this case $\lambda = 3$ .

(a) $P(0 < X < 1)$

P(0 < X < 1) = e^{-3 \times 0} - e^{-3 \times 1} = 1 - e^{-3} \approx 0.9502

(b) $P(0 < X < 3)$

P(0 < X < 3) = e^{0} - e^{-9} = 1 - e^{-9} \approx 0.9999

(c) $P(0 < X < 5)$

P(0 < X < 5) = e^{0} - e^{-15} = 1 - e^{-15} \approx 1

(d) $P(2 < X < 5)$

P(2 < X < 5) = e^{-6} - e^{-15} \approx 0.00248

(e) $P(2 < X < 10)$

P(2 < X < 10) = e^{-6} - e^{-30} \approx 0.00248

(f) $P(X > 2)$

From Example 2.4.5:

P(X \geq x) = e^{-\lambda x}

P(X > 2) = e^{-3 \times 2} = e^{-6} \approx 0.00248