If a widget's reliability is represented by an exponential distribution with beta = 75 (weeks). What is the 50% life expectancy – e.g. the length of time where the reliability = 50%?
To find the 50% life expectancy of a widget with the given exponential distribution, we can use the formula for the mean (or average) of an exponential distribution.
In this case, the mean (μ) of an exponential distribution is equal to the reciprocal of the beta value (β). So, the mean of the given distribution is 1/75 weeks.
Since the exponential distribution is continuous, the probability density function (PDF) is given by:
f(x) = (1 / β) * e^(-x/β)
Where:
- f(x) is the probability density function at a given value x.
- β is the scale parameter of the distribution.
The cumulative distribution function (CDF) of the exponential distribution, which gives the probability that a random variable X is less than or equal to a certain value x, is given by:
F(x) = 1 - e^(-x/β)
The 50% life expectancy corresponds to the value of x where the cumulative distribution function (CDF) equals 0.5. So, we can set F(x) = 0.5 and solve for x.
0.5 = 1 - e^(-x/75)
Rearranging the equation:
e^(-x/75) = 0.5
Taking the natural logarithm (ln) of both sides:
ln(e^(-x/75)) = ln(0.5)
Simplifying:
-x/75 = ln(0.5)
Multiply both sides by -75:
x = 75 * ln(0.5)
Calculating:
x ≈ 51.768 weeks
Therefore, the 50% life expectancy or the length of time where the reliability is 50% for the given widget is approximately 51.768 weeks.