Math Problem Statement

Let X be a continuous random variable with PDF f_X(x)= \begin{cases} kx^2, & 0\leq x \leq 1\ 0, & \text{otherwise.} \end{cases}fX​(x)={kx2,0,​0≤x≤1otherwise.​ Find the median of X. Enter the answer correct to four decimal places.

Hint: The median m is the value such that P(X \leq m) = P(X > m) = 0.5P(Xm)=P(X>m)=0.5

Solution

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Math Problem Analysis

Mathematical Concepts

Probability Density Function (PDF)
Continuous Random Variables
Cumulative Distribution Function (CDF)

Formulas

∫[0 to 1] f_X(x) dx = 1 (PDF normalization)
f_X(x) = kx^2
∫[0 to m] 3x^2 dx = 0.5 (Median calculation)

Theorems

Fundamental Theorem of Calculus
Property of Continuous Random Variable Median: P(X ≤ m) = 0.5

Suitable Grade Level

Undergraduate Probability and Statistics