A random variable X is exponentially distributed with a mean of 0.16.
a-1. What is the rate parameter λ? (Round your answer to 3 decimal places.)
a-2. What is the standard deviation of X? (Round your answer to 2 decimal places.)
b. Compute P(X > 0.24). (Round intermediate calculations to at least 4 decimal places and final answer to 4 decimal places.)
c. Compute P(0.09 ≤ X ≤ 0.24). (Round intermediate calculations to at least 4 decimal places and final answer to 4 decimal places.)
a-1. The rate parameter λ is equal to the reciprocal of the mean. So, λ = 1/0.16 = 6.250.
a-2. The standard deviation of an exponentially distributed random variable is equal to the reciprocal of the rate parameter. So, the standard deviation of X is equal to 1/6.250 = 0.160.
b. To compute P(X > 0.24), we can use the exponential probability density function:
P(X > 0.24) = 1 - P(X ≤ 0.24)
= 1 - (1 - e^(-λx))
= e^(-λx)
= e^(-6.250 * 0.24)
≈ 0.0030 (rounded to 4 decimal places).
c. To compute P(0.09 ≤ X ≤ 0.24), we can again use the exponential probability density function:
P(0.09 ≤ X ≤ 0.24) = P(X > 0.09) - P(X > 0.24)
= e^(-λ * 0.09) - e^(-λ * 0.24)
= e^(-6.250 * 0.09) - e^(-6.250 * 0.24)
≈ 0.0669 (rounded to 4 decimal places).
a-1. To find the rate parameter λ of an exponential distribution with mean μ, we use the formula λ = 1/μ. In this case, the given mean is 0.16, so the rate parameter is λ = 1/0.16 = 6.25 (rounded to 3 decimal places).
a-2. The standard deviation (σ) of an exponential distribution with rate parameter λ is given by σ = 1/λ. Since we found λ to be 6.25 in the previous step, the standard deviation is σ = 1/6.25 = 0.16 (rounded to 2 decimal places).
b. To compute P(X > 0.24), we'll use the formula P(X > x) = e^(-λx), where x is the specified value. Plugging in the value x = 0.24 and λ = 6.25 from the previous steps, we get P(X > 0.24) = e^(-6.25 * 0.24). Evaluating this expression, we find P(X > 0.24) ≈ 0.5820 (rounded to 4 decimal places).
c. To compute P(0.09 ≤ X ≤ 0.24), we'll use the formula P(a ≤ X ≤ b) = e^(-λa) - e^(-λb). Plugging in the values a = 0.09, b = 0.24, and λ = 6.25, we get P(0.09 ≤ X ≤ 0.24) = e^(-6.25 * 0.09) - e^(-6.25 * 0.24). Evaluating this expression, we find P(0.09 ≤ X ≤ 0.24) ≈ 0.2077 (rounded to 4 decimal places).
To solve these problems, we need to understand the exponential distribution and its properties. The exponential distribution is a continuous probability distribution that describes the time between events in a Poisson process, where events occur at a constant average rate. The exponential distribution is typically characterized by its rate parameter (λ), which represents the average rate at which events occur.
To solve these problems, we will use the following formulas:
a-1. The rate parameter (λ) can be calculated using the formula λ = 1 / mean.
a-2. The standard deviation of X can be calculated using the formula σ = 1 / λ.
b. To compute P(X > 0.24), we can use the formula P(X > x) = e^(-λx), where x is the value we want to find the probability for.
c. To compute P(0.09 ≤ X ≤ 0.24), we can use the formula P(a ≤ X ≤ b) = e^(-λa) - e^(-λb), where a and b are the lower and upper bounds of the interval.
Let's solve these problems step-by-step:
a-1. To find the rate parameter (λ), we can use the formula λ = 1 / mean.
In this case, the mean is given as 0.16, so λ = 1 / 0.16 = 6.25 (rounded to 3 decimal places).
a-2. To find the standard deviation of X, we can use the formula σ = 1 / λ.
Using the value of λ calculated in the previous step (6.25), we can calculate the standard deviation σ = 1 / 6.25 = 0.16 (rounded to 2 decimal places).
b. To compute P(X > 0.24), we can use the formula P(X > x) = e^(-λx).
Plugging in the value of λ (6.25) and x (0.24), the calculation is as follows:
P(X > 0.24) = e^(-6.25 * 0.24) ≈ 0.4699 (rounded to 4 decimal places).
c. To compute P(0.09 ≤ X ≤ 0.24), we can use the formula P(a ≤ X ≤ b) = e^(-λa) - e^(-λb).
Plugging in the value of λ (6.25) and the lower bound a (0.09) and upper bound b (0.24), the calculation is as follows:
P(0.09 ≤ X ≤ 0.24) = e^(-6.25 * 0.09) - e^(-6.25 * 0.24) ≈ 0.0589 (rounded to 4 decimal places).
I hope this explanation helps you understand how to solve these exponential distribution problems. If you have any further questions, feel free to ask!