Consider the following exponential probability density function. f (x) = 1 5 e-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.)

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