Suppose X has an exponential distribution with mean equal to 13. Determine the following:
(a) Upper P left-parenthesis x greater-than 10 right-parenthesis (Round your answer to 3 decimal places.)
(b) Upper P left-parenthesis x greater-than 20 right-parenthesis (Round your answer to 3 decimal places.)
(c) Upper P left-parenthesis x less-than 30 right-parenthesis (Round your answer to 3 decimal places.)
(d) Find the value of x such that Upper P left-parenthesis Upper X less-than x right-parenthesis equals 0.95. (Round your answer to 2 decimal places.)
To determine the probabilities and find the value of x for the given exponential distribution, we can use the exponential cumulative distribution function (CDF). The CDF of an exponential distribution is defined as:
F(x) = 1 - e^(-λx)
where λ is the rate parameter, and x is the value at which we want to evaluate the CDF.
In this case, we are given that the mean of the exponential distribution is equal to 13. The rate parameter λ is related to the mean by the equation λ = 1/mean.
(a) To find P(X > 10), we need to evaluate 1 - F(10):
λ = 1/13
F(10) = 1 - e^(-10λ)
P(X > 10) ≈ 1 - F(10)
Substituting the values, we can calculate P(X > 10).
(b) Similarly, to find P(X > 20), we need to evaluate 1 - F(20):
λ = 1/13
F(20) = 1 - e^(-20λ)
P(X > 20) ≈ 1 - F(20)
Substituting the values, we can calculate P(X > 20).
(c) To find P(X < 30), we can directly evaluate F(30):
λ = 1/13
F(30) = 1 - e^(-30λ)
P(X < 30) ≈ F(30)
Substituting the values, we can calculate P(X < 30).
(d) To find the value of x such that P(X < x) = 0.95, we need to find the quantile using the inverse of the CDF:
λ = 1/13
P(X < x) = 0.95
0.95 = 1 - e^(-λx)
Rearranging the equation, we can solve for x:
e^(-λx) = 1 - 0.95
e^(-λx) = 0.05
Taking the natural logarithm (ln) on both sides:
-λx = ln(0.05)
x = ln(0.05) / -λ
Substituting the values, we can calculate the value of x that satisfies P(X < x) = 0.95.
Please note that to obtain the final numerical results, you will need to substitute the value of λ (1/13) into the equations and perform the calculations using a calculator or mathematical software.