LED light bulbs are manufactured with an average life span of 16 years and the bulb lifespans are known to follow an exponential distribution.

a. What is the probability that a bulb will last longer than 14 years?
b. What is the 50th percentile (or median) of bulb lifespans?

P (x>14)=e^(-14/16)= 0.4169=41.6%

P(X>/=k)
Solve for K as K will be the median value.

e^(-k/16)=1/2
Shorthand to find the median of an exponential distr. is:
Meanxln(2)=k

meaning 16xln2=11.09
This is the value of a median, not a percentage.

To answer these questions, we can use the properties of the exponential distribution. The exponential distribution is a continuous probability distribution commonly used to model the lifespan of products.

The probability density function (pdf) of the exponential distribution is given by:

f(x) = (1/μ) * exp(-x/μ)

where μ is the average lifespan of the bulb.

a. To find the probability that a bulb will last longer than 14 years, we need to calculate the complementary cumulative distribution function (ccdf), which is the integral of the pdf from 14 years to positive infinity.

P(X > 14) = ∫[14, ∞] (1/μ) * exp(-x/μ) dx

To solve the integral, we can integrate by parts or use a table of integrals. However, in this case, we can use the complementary property of the exponential distribution. The ccdf of an exponential distribution is given by:

P(X > x) = 1 - P(X ≤ x) = 1 - exp(-x/μ)

Therefore, the probability that a bulb will last longer than 14 years is:

P(X > 14) = 1 - exp(-14/16) = 1 - exp(-0.875) ≈ 0.3956 (or 39.56%)

b. The 50th percentile (median) of bulb lifespans is the value below which 50% of the bulb lifespans fall. In other words, it is the value x for which P(X ≤ x) = 0.5.

Therefore, we need to solve the equation:

0.5 = P(X ≤ x) = 1 - P(X > x) = 1 - (1 - exp(-x/μ))

Simplifying this equation, we have:

0.5 = exp(-x/μ)

To solve for the median, we need to find the value of x that satisfies this equation. We can take the natural logarithm of both sides and solve for x:

ln(0.5) = -x/μ

x = -μ * ln(0.5)

Substituting the average lifespan μ = 16 years, we can calculate the median:

x = -16 * ln(0.5) ≈ 11.5129 years

Therefore, the 50th percentile (median) of bulb lifespans is approximately 11.5129 years.