The life of automobile voltage regulators has an exponential distribution with a mean life of six years. You purchase an automobile that is six years old, with a working voltage regulator and plan to own it for three years.
a)What is the probability that the voltage regulator fails during your ownership?
Pr(x>9)=.223
Pr(fails)=.5-.223
So the number of fails is 4.777? How id you get .5?
To find the probability that the voltage regulator fails during your ownership, we can use the exponential distribution.
The exponential distribution is defined by the equation:
f(x;λ) = λ * exp(-λx)
Where λ is the rate parameter, which is equal to the inverse of the mean (1/mean).
In this case, the mean life of the voltage regulator is given as 6 years. So the rate parameter (λ) is 1/6.
To find the probability of failure during your ownership (in 3 years), we can calculate the cumulative distribution function (CDF) at 3 years.
The CDF of the exponential distribution is defined as:
F(x;λ) = 1 - exp(-λx)
So, for our case, we can calculate the probability of failure during your ownership as:
P(failure) = 1 - exp(-(1/6) * 3)
Now, let's calculate the probability:
P(failure) = 1 - exp(-(1/6) * 3)
P(failure) = 1 - exp(-1/2)
P(failure) ≈ 0.3935
Therefore, the probability that the voltage regulator fails during your ownership is approximately 0.3935 or 39.35%.
To find the probability that the voltage regulator fails during your ownership, we can use the exponential distribution formula.
The exponential distribution formula is given by:
f(x) = λ * exp(-λx)
Where:
- f(x) is the probability density function (pdf)
- λ is the rate parameter (λ = 1/mean)
- exp is the exponential function
- x is the time
Based on the given information, we know that the mean life of automobile voltage regulators is six years. Hence, the rate parameter λ can be calculated as follows:
λ = 1/mean = 1/6 years^-1 = 1/6
Now, we need to find the probability of the voltage regulator failing during a three-year ownership. Let's denote this time as t.
To find this probability, we need to integrate the probability density function from t to infinity. In other words, we need to calculate the integral of f(x) from t to infinity.
P(voltage regulator fails during ownership) = ∫(t to ∞) λ * exp(-λx) dx
Since we are integrating from t to infinity, the upper limit can be approximated as a very large number. In practice, we can use a large value such as 10,000.
P(voltage regulator fails during ownership) = ∫(t to 10000) λ * exp(-λx) dx
To solve this integral, we can use integration techniques or numerical methods such as software or calculators.