Suppose X has an exponential distribution with . Determine the following. Round the answers to 3 decimal places.
a) P(x<=0)
b) P(X>=2)
c) P(X<=1)
d) P(1<x<2)
e) Find the value of x such that P(X<x)=.05
To determine the probabilities in this case, we need to use the probability density function (PDF) of the exponential distribution. The PDF of an exponential distribution is given by:
f(x) = λ * e^(-λx), for x >= 0
Where:
- λ (lambda) is the rate parameter of the exponential distribution.
Given the rate parameter, we can then calculate the probabilities. Let's go through the calculations step by step:
a) P(x <= 0):
To calculate this probability, we need to integrate the PDF from 0 to infinity. However, since the exponential distribution is only defined for positive values of x, P(x <= 0) is equal to 0.
b) P(X >= 2):
Here, we need to calculate the complement of the probability, which means finding P(X < 2) and subtracting it from 1. To calculate P(X < 2), we integrate the PDF from 0 to 2. Let's denote the integral as I(0, 2):
I(0, 2) = ∫[0 to 2] λ * e^(-λx) dx
To calculate this integral, we can use the following indefinite integral:
∫ e^(-λx) dx = (-1/λ) * e^(-λx) + C
Applying the limits of integration, we have:
I(0, 2) = (-1/λ) * e^(-λx) from 0 to 2
To find P(X >= 2), we subtract I(0, 2) from 1:
P(X >= 2) = 1 - I(0, 2)
c) P(X <= 1):
To calculate this probability, we integrate the PDF from 0 to 1:
I(0, 1) = ∫[0 to 1] λ * e^(-λx) dx
Similarly to the previous calculation, we use the indefinite integral:
∫ e^(-λx) dx = (-1/λ) * e^(-λx) + C
Applying the limits of integration, we get:
I(0, 1) = (-1/λ) * e^(-λx) from 0 to 1
d) P(1 < x < 2):
To calculate this probability, we need to find the difference between P(X < 2) and P(X < 1):
P(1 < x < 2) = P(X < 2) - P(X < 1)
e) Find the value of x such that P(X < x) = 0.05:
To find the value of x, we need to find the inverse of the cumulative distribution function (CDF) of the exponential distribution. The CDF of the exponential distribution is given by:
F(x) = 1 - e^(-λx)
We set F(x) equal to 0.05 and solve for x:
0.05 = 1 - e^(-λx)
e^(-λx) = 0.95
Taking the natural logarithm (ln) of both sides:
-ln(0.95) = -λx
Solving for x:
x = -ln(0.95) / λ
This value of x gives P(X < x) = 0.05.
Note: In order to provide specific numerical answers, we need to know the value of λ (the rate parameter).