(a) To write the formula for P(x < x_o), we need to calculate the cumulative distribution function (CDF) of the given exponential probability density function (PDF). The CDF represents the probability that the random variable is less than or equal to a certain value.
For the exponential distribution with parameter λ = 1/4, the CDF is given by:
F(x) = 1 - e^(-λx) for x ≥ 0
Therefore, the formula for P(x < x_o) is:
P(x < x_o) = F(x_o) = 1 - e^(-λx_o)
(b) To find P(x ≥ 3), we can use the CDF and subtract it from 1 since it represents the probability of the random variable being greater than or equal to a certain value.
P(x ≥ 3) = 1 - F(3) = 1 - (1 - e^(-λ(3))) = e^(-λ(3))
Substituting λ = 1/4:
P(x ≥ 3) = e^(-1/4 * 3) = e^(-3/4)
(c) To find P(x < 4), we can use the CDF:
P(x < 4) = F(4) = 1 - e^(-λ(4))
Substituting λ = 1/4:
P(x < 4) = 1 - e^(-1/4 * 4) = 1 - e^(-1)
(d) To find P(x > 5), we can subtract the CDF of 5 from 1:
P(x > 5) = 1 - F(5) = 1 - (1 - e^(-λ(5))) = e^(-λ(5))
Substituting λ = 1/4:
P(x > 5) = e^(-1/4 * 5) = e^(-5/4)
(e) To find P(3 ≤ x ≤ 5), we subtract the CDF of 3 from the CDF of 5:
P(3 ≤ x ≤ 5) = F(5) - F(3) = (1 - e^(-λ(5))) - (1 - e^(-λ(3))) = e^(-λ(3)) - e^(-λ(5))
Substituting λ = 1/4:
P(3 ≤ x ≤ 5) = e^(-1/4 * 3) - e^(-1/4 * 5) = e^(-3/4) - e^(-5/4)
Remember to round your answers to four decimal places as requested.