A random variable X is exponentially distributed with an expected value of 50.
a-1. What is the rate parameter λ? (Round your answer to 3 decimal places.)
a-2. What is the standard deviation of X?
b. Compute P(44 ≤ X ≤ 56). (Round intermediate calculations to at least 4 decimal places and final answer to 4 decimal places.)
c. Compute P(36 ≤ X ≤ 64). (Round intermediate calculations to at least 4 decimal places and final answer to 4 decimal places.)
a-1. To find the rate parameter λ, we use the formula for the expected value of an exponentially distributed variable:
E(X) = 1/λ
Given that the expected value E(X) is 50, we can solve for λ:
50 = 1/λ
λ = 1/50 = 0.02 (rounded to 3 decimal places)
Therefore, the rate parameter λ is 0.02.
a-2. The standard deviation of an exponentially distributed variable with rate parameter λ is given by the formula:
σ = 1/λ
Using the rate parameter λ = 0.02 we found in part a-1:
σ = 1/0.02 = 50
Therefore, the standard deviation of X is 50.
b. To compute P(44 ≤ X ≤ 56), we use the cumulative distribution function (CDF) of the exponential distribution. The CDF is given by:
CDF(x) = 1 - e^(-λx)
P(44 ≤ X ≤ 56) = CDF(56) - CDF(44)
= (1 - e^(-0.02*56)) - (1 - e^(-0.02*44))
= (1 - e^(-1.12)) - (1 - e^(-0.88))
≈ 0.4970
Therefore, P(44 ≤ X ≤ 56) is approximately 0.4970.
c. Similarly, to compute P(36 ≤ X ≤ 64), we use the CDF:
P(36 ≤ X ≤ 64) = CDF(64) - CDF(36)
= (1 - e^(-0.02*64)) - (1 - e^(-0.02*36))
= (1 - e^(-1.28)) - (1 - e^(-0.72))
≈ 0.5042
Therefore, P(36 ≤ X ≤ 64) is approximately 0.5042.
a-1. To find the rate parameter λ of an exponentially distributed random variable, we can use the formula:
λ = 1 / E(X)
where E(X) is the expected value of X. Given that the expected value of X is 50, we can calculate the rate parameter as follows:
λ = 1 / 50 ≈ 0.02 (rounded to 3 decimal places)
Therefore, the rate parameter λ is approximately 0.02.
a-2. The standard deviation of an exponentially distributed random variable is given by the formula:
σ = 1 / λ
where λ is the rate parameter. Using the value of λ calculated in part a-1, we can calculate the standard deviation of X as follows:
σ = 1 / 0.02 = 50
Therefore, the standard deviation of X is 50.
b. To compute P(44 ≤ X ≤ 56), we need to use the cumulative distribution function (CDF) of the exponential distribution. The CDF of an exponential distribution is given by the formula:
CDF(x) = 1 - e^(-λx)
Using the rate parameter λ calculated in part a-1, we can substitute the values into the formula and calculate the probabilities as follows:
P(44 ≤ X ≤ 56) = CDF(56) - CDF(44)
= (1 - e^(-0.02 * 56)) - (1 - e^(-0.02 * 44))
Calculating the probabilities using a calculator or software, we find:
P(44 ≤ X ≤ 56) ≈ 0.1847 (rounded to 4 decimal places)
Therefore, the probability that X falls between 44 and 56 is approximately 0.1847.
c. To compute P(36 ≤ X ≤ 64), we can use the same CDF formula and substitute the appropriate values:
P(36 ≤ X ≤ 64) = CDF(64) - CDF(36)
= (1 - e^(-0.02 * 64)) - (1 - e^(-0.02 * 36))
Calculating the probabilities using a calculator or software, we find:
P(36 ≤ X ≤ 64) ≈ 0.6651 (rounded to 4 decimal places)
Therefore, the probability that X falls between 36 and 64 is approximately 0.6651.
To find the rate parameter λ, we need to use the formula for the expected value of an exponentially distributed random variable: E(X) = 1/λ. Given that the expected value (E(X)) is 50, we can solve for λ.
a-1. To find the rate parameter λ, we use the formula E(X) = 1/λ.
50 = 1/λ
To solve for λ, we need to isolate it on one side of the equation.
λ = 1/50
Calculating, we find that the rate parameter λ is 0.02 (rounded to 3 decimal places).
To find the standard deviation of X, we can use the formula for the standard deviation of an exponentially distributed random variable: σ = 1/λ.
a-2. To find the standard deviation (σ), we use the formula σ = 1/λ.
σ = 1/0.02
Calculating, we find that the standard deviation of X is 50 (rounded to the nearest whole number).
To compute probabilities for X within a certain range, we need to use the cumulative distribution function (CDF) of the exponential distribution. For the exponential distribution, the CDF is given by the formula P(X ≤ x) = 1 - e^(-λx).
b. To compute P(44 ≤ X ≤ 56), we need to find P(X ≤ 56) and subtract P(X ≤ 44) from it.
First, we find P(X ≤ 44) using the CDF formula:
P(X ≤ 44) = 1 - e^(-0.02 * 44)
Calculating, we get P(X ≤ 44) ≈ 0.4996 (rounded to 4 decimal places).
Next, we find P(X ≤ 56) using the CDF formula:
P(X ≤ 56) = 1 - e^(-0.02 * 56)
Calculating, we get P(X ≤ 56) ≈ 0.6321 (rounded to 4 decimal places).
To find P(44 ≤ X ≤ 56), we subtract P(X ≤ 44) from P(X ≤ 56):
P(44 ≤ X ≤ 56) = P(X ≤ 56) - P(X ≤ 44)
P(44 ≤ X ≤ 56) ≈ 0.6321 - 0.4996
Calculating, we get P(44 ≤ X ≤ 56) ≈ 0.1325 (rounded to 4 decimal places).
c. To compute P(36 ≤ X ≤ 64), we follow the same steps as in part b.
First, we find P(X ≤ 36) using the CDF formula:
P(X ≤ 36) = 1 - e^(-0.02 * 36)
Calculating, we get P(X ≤ 36) ≈ 0.2835 (rounded to 4 decimal places).
Next, we find P(X ≤ 64) using the CDF formula:
P(X ≤ 64) = 1 - e^(-0.02 * 64)
Calculating, we get P(X ≤ 64) ≈ 0.7726 (rounded to 4 decimal places).
To find P(36 ≤ X ≤ 64), we subtract P(X ≤ 36) from P(X ≤ 64):
P(36 ≤ X ≤ 64) = P(X ≤ 64) - P(X ≤ 36)
P(36 ≤ X ≤ 64) ≈ 0.7726 - 0.2835
Calculating, we get P(36 ≤ X ≤ 64) ≈ 0.4891 (rounded to 4 decimal places).