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?

a) P(x<3) = F(3)

= 1-e^((-1/6)(3))
=0.3934693403

b) E(X) = 6

Well, let's start by plugging in the given information into our mathematical equation... Oh wait, wrong script! Let me try that again.

So we have an exponential distribution with a mean life of six years. That means the average lifespan of a voltage regulator is six years.

Since you purchased a six-year-old automobile, the voltage regulator has already survived its average lifespan. So if it has made it this far, it's like a superhero with bonus powers.

Now, you plan on owning the car for three years. Since the voltage regulator has already been around for six years, it only needs to survive these three additional years to avoid being a disappointment.

The exponential distribution is memoryless, so the past doesn't affect the future. Therefore, the probability that the voltage regulator fails during your ownership is the same as the probability that it fails during any three-year period.

To calculate this, we'll use the exponential decay formula: P(X > t) = e^(-t/μ), where μ is the mean lifetime.

Plugging in the values, we get: P(X > 3) = e^(-3/6)

Now, let's do some math! Grab your calculators or brains if you're feeling adventurous.

Calculating... calculating... *insert clownish music* The probability that the voltage regulator fails during your ownership is approximately 0.6065, or 60.65%.

So there you go, there's a 60.65% chance for some shocking surprises during your ownership! Just kidding. Good luck with your automobile!

To find the probability that the voltage regulator fails during your ownership, we can use the exponential distribution.

The exponential distribution is defined by its rate parameter (λ), which is the reciprocal of the mean (μ). In this case, the mean life of the voltage regulators is given as six years, so the rate parameter can be calculated as λ = 1/6.

Let's denote the random variable representing the lifetime of the voltage regulator as X. Since X follows an exponential distribution with rate parameter λ, we can write X ~ Exp(1/6).

Now, we want to find the probability that the voltage regulator fails during your ownership, which is equivalent to finding P(X ≤ 3).

The cumulative distribution function (CDF) of the exponential distribution is given by P(X ≤ x) = 1 - e^(-λx).

Substituting the values, we have P(X ≤ 3) = 1 - e^(-1/6 * 3).

Calculating this, we get:

P(X ≤ 3) = 1 - e^(-1/2) ≈ 0.39347

Therefore, the probability that the voltage regulator fails during your ownership is approximately 0.39347 (or 39.35%).

To calculate the probability that the voltage regulator fails during your ownership, we can use the exponential distribution formula.

The exponential distribution has a probability density function (PDF) given by:
f(x) = λ * e^(-λx)

Where:
- λ is the rate parameter, which is equal to the inverse of the mean (λ = 1/mean)
- e is the mathematical constant approximately equal to 2.71828
- x is the time period

In this case, the mean life of the voltage regulators is given as 6 years, so λ = 1/6.

Now, we want to find the probability that the voltage regulator fails during your ownership period of 3 years. Let's call this event A.

To find the probability of event A occurring, we need to calculate the cumulative distribution function (CDF) of the exponential distribution up to the value of 3 years. The CDF gives the probability that the random variable is less than or equal to a particular value.

The CDF of the exponential distribution is given by:
F(x) = 1 - e^(-λx)

Substituting the values, we have:
F(3) = 1 - e^(-1/6 * 3)

Calculating this, we find:
F(3) = 1 - e^(-1/2) ≈ 0.3935

Therefore, the probability that the voltage regulator fails during your ownership is approximately 0.3935, or 39.35%.