Question
A random variable X has the following probability distribution:
f ( x) = {cx2 for 0< x < 3 0 otherwise
Determine the value of the constant c and Var (X).
Answer :
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Skip in 10
To determine the value of the constant \( c \) and the variance \( \text{Var}(X) \), we proceed step by step. ### Step 1: Find \( c \) The given probability density function (PDF) is: \[ f(x) = c x^2, \quad 0 < x < 3 \] Since \( f(x) \) is a probability density function, it must satisfy the normalization condition: \[ \int_{0}^{3} f(x) \, dx = 1 \] Substituting \( f(x) = c x^2 \): \[ \int_{0}^{3} c x^2 \, dx = 1 \] Evaluating the integral: \[ c \int_{0}^{3} x^2 \, dx = 1 \] \[ c \left[ \frac{x^3}{3} \right]_{0}^{3} = 1 \] \[ c \left( \frac{3^3}{3} - \frac{0^3}{3} \right) = 1 \] \[ c \left( \frac{27}{3} \right) = 1 \] \[ c \times 9 = 1 \] \[ c = \frac{1}{9} \] ### Step 2: Find \( E(X) \) The expectation
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