The life of an electric component has an exponential distribution with a mean of 10 years. What is the probability that a randomly selected one such component has a life more than 7 years?
Need to know standard deviation to calculate probability.
To find the probability that a randomly selected electric component has a life more than 7 years, we can use the exponential distribution formula.
The exponential distribution probability density function is given by:
f(x) = λ * e^(-λx)
where λ is the rate parameter.
In this case, the mean (μ) is 10 years. Since the mean of the exponential distribution is equal to 1/λ, we can calculate the rate parameter (λ) as:
λ = 1 / μ
λ = 1 / 10
λ = 0.1
Now, we want to find the probability that the component has a life more than 7 years, which is equivalent to finding the probability that the component survives beyond 7 years. We can express this as:
P(X > 7)
Using the exponential distribution cumulative distribution function (CDF), we can calculate this probability as:
P(X > 7) = 1 - F(7)
where F(x) is the CDF.
The exponential distribution CDF is given by:
F(x) = 1 - e^(-λx)
Substituting the values into the equation, we have:
P(X > 7) = 1 - F(7)
P(X > 7) = 1 - (1 - e^(-0.1 * 7))
P(X > 7) = 1 - (1 - e^(-0.7))
P(X > 7) = 1 - (1 - 0.4966)
P(X > 7) = 0.4966
Therefore, the probability that a randomly selected electric component has a life more than 7 years is approximately 0.4966 or 49.66%.
To find the probability that a randomly selected electric component has a life more than 7 years, we need to calculate the cumulative distribution function (CDF) of the exponential distribution and then subtract it from 1.
The exponential distribution has the probability density function (PDF) given by:
f(x) = λ * e^(-λx)
where λ is the rate parameter, which is the reciprocal of the mean (λ = 1/10 in this case).
To calculate the cumulative distribution function (CDF), we integrate the PDF from 0 to x:
F(x) = ∫[0 to x] f(t) dt = ∫[0 to x] λ * e^(-λt) dt
Now, let's calculate the CDF:
F(x) = ∫[0 to x] λ * e^(-λt) dt = [-e^(-λt)] [0 to x] = -e^(-λx) + e^(-λ*0) = -e^(-λx) + 1
To find the probability that a randomly selected component has a life more than 7 years, we need to find P(X > 7), where X is a random variable representing the component's life.
P(X > 7) = 1 - P(X ≤ 7) = 1 - F(7)
Plugging in the values:
P(X > 7) = 1 - (-e^(-λ*7) + 1) = 1 + e^(-λ*7) - 1 = e^(-λ*7)
Since λ = 1/10, the probability that a randomly selected electric component has a life more than 7 years is:
P(X > 7) = e^(-1/10 * 7) = e^(-7/10) ≈ 0.4966
Therefore, the probability is approximately 0.4966 or 49.66%.