The Probability density function for a continuous random variable X is given by 𝑓(𝑥)=𝑥34 for 0<x<2 and 0 elsewhere. What is the cumulative distribution function of X?
Question
The Probability density function for a continuous random variable X is given by 𝑓(𝑥)=𝑥34 for 0<x<2 and 0 elsewhere. What is the cumulative distribution function of X?
Solution
The cumulative distribution function (CDF) of a random variable X is defined as the probability that X will take a value less than or equal to x. It is calculated by integrating the probability density function (PDF) from the lower limit to x.
Given the PDF f(x) = x/4 for 0 < x < 2 and 0 elsewhere, we can calculate the CDF F(x) as follows:
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For x < 0 or x > 2, the CDF F(x) = 0 because the PDF is 0 outside the interval (0, 2).
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For 0 < x < 2, we integrate the PDF from 0 to x:
∫f(t) dt from 0 to x = ∫(t/4) dt from 0 to x = [t^2/8] from 0 to x = x^2/8.
So, the CDF F(x) is given by:
F(x) = 0 for x ≤ 0 or x ≥ 2, F(x) = x^2/8 for 0 < x < 2.
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