SOLUTION: The time X in months until failure of a certain product has a probability density function of https://i.gyazo.com/8c054188451da18fcdc1c3919fea6bf2.png
for x > 0 and 0 otherwise. F
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-> SOLUTION: The time X in months until failure of a certain product has a probability density function of https://i.gyazo.com/8c054188451da18fcdc1c3919fea6bf2.png
for x > 0 and 0 otherwise. F
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Question 1136622: The time X in months until failure of a certain product has a probability density function of https://i.gyazo.com/8c054188451da18fcdc1c3919fea6bf2.png
for x > 0 and 0 otherwise. Find the cumulative distribution and the median
value of X. Answer by jim_thompson5910(35256) (Show Source):
You can put this solution on YOUR website! I'll break this into two parts. Part A will show how to find the cumulative distribution function (CDF) while part B shows how to find the median. Part B will use the result from part A.
Part A
To find the CDF, we apply the integral to the probability distribution function (PDF) like so
From here, we apply u-substitution.
Let u = (-x/4)^3, so
du = -1*3(1/4)*(-x/4)^2dx
du = -(3/4)*(-x/4)^2dx
du = -(3/4)*((x^2)/16)dx
du = (-3x^2dx)/64
64du = -3x^2*dx
3x^2*dx = -64du
Replace the (-x/4)^3 exponent with 'u'; replace the 3x^2dx with -64du
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Let g(x) = -e^(-(x/4)^3)+C
The area under the f(x) curve is exactly equal to g(k) - g(0) where k approaches positive infinity
In other words,
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Let's see what happens when k approaches positive infinity
So we get an area of 1 as expected, indicating that is the proper CDF for the PDF
note: g(k) = P(x < k) meaning that the CDF returns the area under the f(x) curve to the left of the x = k value.
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