given an exponential distribution with mean = 30 what is the probability that the arrival time is a) less than X = 0.1 B) greater than X = 0.1 c) between X=0.1 and X =0.2 d) less than X =0.01 or greater than X = 0.2
To solve these probability calculations for an exponential distribution with mean = 30, we can use the cumulative distribution function (CDF) of the exponential distribution.
The cumulative distribution function (CDF) for an exponential distribution is given by:
CDF(x) = 1 - e^(-λx)
Where:
- λ (lambda) is the rate parameter, which is equal to 1 / mean in the case of exponential distribution.
Given that mean = 30, we can find the rate parameter:
λ = 1 / mean = 1 / 30
Now, let's calculate the probabilities step by step:
a) Probability that the arrival time is less than X = 0.1
To find this probability, we plug in X = 0.1 into the CDF equation:
CDF(0.1) = 1 - e^(-λ * 0.1)
Substituting λ = 1 / 30:
CDF(0.1) = 1 - e^(-(1/30) * 0.1)
Now, calculate the CDF(0.1) to find the probability.
b) Probability that the arrival time is greater than X = 0.1
Since the exponential distribution is continuous, the probability of a specific value is zero. Therefore, the probability that the arrival time is greater than X = 0.1 is equal to 1 minus the probability that it is less than or equal to X = 0.1.
P(X > 0.1) = 1 - CDF(0.1)
c) Probability that the arrival time is between X = 0.1 and X = 0.2
To find this probability, we subtract the CDF values at X = 0.2 and X = 0.1:
P(0.1 < X < 0.2) = CDF(0.2) - CDF(0.1)
d) Probability that the arrival time is less than X = 0.01 or greater than X = 0.2
To find this probability, we add the probabilities of the arrival time being less than X = 0.01 and greater than X = 0.2:
P(X < 0.01 or X > 0.2) = CDF(0.01) + (1 - CDF(0.2))
Using the above steps, you can calculate the specific probabilities for your exponential distribution with mean = 30.
To find the probabilities for the given exponential distribution, we can use the cumulative distribution function (CDF) of the exponential distribution. The CDF of an exponential distribution is given by:
CDF(x) = 1 - e^(-λx)
where λ is the rate parameter, which is equal to 1/mean for the exponential distribution.
Given that the mean (μ) of the exponential distribution is 30, we can find the rate parameter (λ) as λ = 1/μ = 1/30.
a) Probability that the arrival time is less than X = 0.1:
To find this probability, we need to calculate the CDF value at x = 0.1.
CDF(0.1) = 1 - e^(-λ * 0.1)
Substituting the value of λ, we get:
CDF(0.1) = 1 - e^(-1/30 * 0.1)
Now you can calculate this value using a scientific calculator.
b) Probability that the arrival time is greater than X = 0.1:
To find this probability, we subtract the probability found in part a) from 1.
P(X > 0.1) = 1 - CDF(0.1)
c) Probability that the arrival time is between X = 0.1 and X = 0.2:
To find this probability, we subtract the CDF at x = 0.2 from the CDF at x = 0.1.
P(0.1 < X < 0.2) = CDF(0.2) - CDF(0.1)
d) Probability that the arrival time is less than X = 0.01 or greater than X = 0.2:
To find this probability, we add the probability of X < 0.01 to the probability of X > 0.2.
P(X < 0.01 or X > 0.2) = CDF(0.01) + (1 - CDF(0.2))
By using the formulas above and substituting the values for λ and the given values of x, you can find the respective probabilities.