Question
Download Solution PDFThe probability density function of a random variable X is f(x) = \(\frac{\pi}{10} sin \frac{\pi x}{5}\); 0 ≤ x ≤ 5. The first quartile of X is:
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFCalculation:
To find the first quartile of X, we need to find the value x such that the area under the probability density function f(x) to the left of x is 0.25.
We can begin by finding the cumulative distribution function (CDF) of X:
F(x) = ∫[0, x] f(t) dt
Since f(x) is a uniform distribution between 0 and 5, we have:
f(x) = 1/5 for 0 ≤ x ≤ 5
Therefore, the CDF of X is:
F(x) = ∫[0, x] (1/5) dt = x/5
To find the first quartile of X, we need to find the value x such that:
F(x) = x/5 = 0.25
Solving for x, we get:
x = 0.25 × 5 = 1.25
Therefore, the first quartile of X is 1.25.
Last updated on Jul 9, 2025
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