Question

Find the 25 th percentile of the distribution having pdf \(f(x)=|x| / 4\), where \(-2

Short answer

The 25th percentile of the distribution is \(x = -2\)

Step-by-step solution

Find the Cumulative Distribution Function (CDF)

The CDF is the integral of the PDF here i.e., the area under the pdf curve from -2 to x. Here, since the pdf \(f(x)\) is defined by \(f(x)=|x| / 4\) and for -2 < x < 2, you have to integrate \(f(x)\) from -2 to x. This results in the following CDF: \(F(x) = \int_{-2}^{x} f(x') dx' = - \int_{-2}^{0} (x' / 4) dx' + \int_{0}^{x} (x' / 4) dx' = 1/2 + x^2 / 16\), for \(0 <= x <= 2\)

Set CDF equal to 0.25

The 25th percentile is the point at which 25% (or 0.25) of the data is below. So, you have to set \(F(x)\) to 0.25 and solve for x. This gives: \(1/2 + x^2 / 16 = 0.25\).

Solve the equation for x

Solving the equation for x gets you the value of the 25th percentile. Firstly, rewrite the equation: \( x^2 / 16 = 0.25 - 0.5 = -0.25 \). To find x, multiply both sides by 16: \( x^2 = -4 \). The square root of -4 is not real, which means our assumption that the percentile lies in 0 <= x <= 2 was wrong. The correct range is -2 <= x < 0. For this range, \(F(x) = 1/2 - x^2 / 16 = 0.25\). Solving this equation for x gives: \( x^2 = 4\), and \( x = -2\).

Final Answer

Since the percentile falls in the range -2 <= x < 0, the 25th percentile of the distribution is x = -2