Answer:
So, to summarize:
a. Probability of failing during the first 100 hours ≈ 0.3220
b. Probability of failing between 50 and 150 hours ≈ 0.1318
c. Probability of surviving more than 200 hours ≈ 0.6780
d. Expected time to failure ≈ 250 hours
e. Variance ≈ 62500 hours^2, Standard Deviation ≈ 250 hours
Explanation:
To solve these problems, we'll use the exponential distribution with the given failure rate of 0.004 tire per hour. The probability density function (PDF) for the exponential distribution is given by:
f(x) = λ * e^(-λx)
Where:
λ (lambda) is the failure rate (0.004 tire per hour).
x is the time to failure (in hours).
e is the base of the natural logarithm, approximately 2.71828.
Let's address each part of the problem:
a. Probability of failing during the first 100 hours (0 ≤ x ≤ 100):
To find this probability, we need to calculate the integral of the PDF from 0 to 100:
P(0 ≤ x ≤ 100) = ∫[0 to 100] λ * e^(-λx) dx
P(0 ≤ x ≤ 100) = λ * ∫[0 to 100] e^(-λx) dx
P(0 ≤ x ≤ 100) = λ * [-e^(-λx)] [from 0 to 100]
P(0 ≤ x ≤ 100) = λ * [-e^(-100λ) + 1]
P(0 ≤ x ≤ 100) = 0.004 * [-e^(-0.4) + 1]
P(0 ≤ x ≤ 100) ≈ 0.3220
b. Probability of failing between 50 and 150 hours (50 ≤ x ≤ 150):
Similarly, we calculate the integral of the PDF from 50 to 150:
P(50 ≤ x ≤ 150) = ∫[50 to 150] λ * e^(-λx) dx
P(50 ≤ x ≤ 150) = λ * ∫[50 to 150] e^(-λx) dx
P(50 ≤ x ≤ 150) = λ * [-e^(-λx)] [from 50 to 150]
P(50 ≤ x ≤ 150) = λ * [-e^(-150λ) + e^(-50λ)]
P(50 ≤ x ≤ 150) = 0.004 * [-e^(-0.6) + e^(-0.2)]
P(50 ≤ x ≤ 150) ≈ 0.1318
c. Probability of surviving more than 200 hours (x > 200):
This is the complement of the probability of failing in the first 200 hours:
P(x > 200) = 1 - [P(0 ≤ x ≤ 200)]
P(x > 200) = 1 - [0.3220]
P(x > 200) ≈ 0.6780
d. Expected time to failure (mean or average):
The expected time to failure for a random tire is given by the inverse of the failure rate (lambda):
Expected Time to Failure = 1 / λ
Expected Time to Failure = 1 / 0.004
Expected Time to Failure = 250 hours
and. Variance and Standard Deviation:
The variance (σ^2) of the exponential distribution is given by:
Variance (σ^2) = 1 / λ^2
Variance (p^2)
Variance (σ^2) = 62500
The standard deviation (σ) is the square root of the variance:
Standard Deviation (σ) = √(62500)
Standard Deviation (σ) = 250 hours