Question:
Consider the following exponential probability density function.
f(x) = 1/5e^(−x/5) for x ≥ 0
(a) Write the formula for P(x ≤ x0). (b) Find P(x ≤ 4). (Round your answer to four decimal places.) (c) Find P(x ≥ 5). (Round your answer to four decimal places.) (d) Find P(x ≤ 6). (Round your answer to four decimal places.) (e) Find P(4 ≤ x ≤ 6). (Round your answer to four decimal places.)
Answer:
(a) P(x ≤ x0) = 1 - e^(−x0/5)
(b) P(x ≤ 4) = 0.5506
(c) P(x ≥ 5) = 0.3678
(d) P(x ≤ 6) = 0.6988
(e) P(4 ≤ x ≤ 6) = 0.1482
Step-by-step explanation:
The standard form of the exponential probability density function is given by
f(x) = 1/μe^(−x/μ)
Where μ is the mean, for the given problem μ = 5
(a) Write the formula for P(x ≤ x0)
P(x ≤ x0) = 1 - e^(−x0/5)
(b) Find P(x ≤ 4)
P(x ≤ 4) = 1 - e^(−4/5)
P(x ≤ 4) = 1 - 0.4493
P(x ≤ 4) = 0.5506
(c) Find P(x ≥ 5)
P(x ≥ 5) = e^(−5/5)
P(x ≥ 5) = 0.3678
(d) Find P(x ≤ 6)
P(x ≤ 6) = 1 - e^(−6/5)
P(x ≤ 6) = 1 - 0.3011
P(x ≤ 6) = 0.6988
(e) Find P(4 ≤ x ≤ 6)
P(4 ≤ x ≤ 6) = e^(−4/5) - e^(−6/5)
P(4 ≤ x ≤ 6) = 0.4493 - 0.3011
P(4 ≤ x ≤ 6) = 0.1482
