Find k such that the function f defined byfX (x) =kx2, 0 < x < 1,0, otherwiseis a probability density function. Also, determine P (1/3 < X ≤ 1/2
Question
Find k such that the function f defined byfX (x) =kx2, 0 < x < 1,0, otherwiseis a probability density function. Also, determine P (1/3 < X ≤ 1/2
Solution
To find the value of k such that the function f is a probability density function, we need to ensure that the integral of f from -∞ to ∞ is equal to 1, because the total probability must be 1.
The function f is only non-zero between 0 and 1, so we only need to integrate from 0 to 1:
∫ from 0 to 1 of kx^2 dx = [kx^3 / 3] from 0 to 1 = k/3 - 0 = k/3.
Setting this equal to 1 gives k/3 = 1, so k = 3.
To find P(1/3 < X ≤ 1/2), we integrate the probability density function from 1/3 to 1/2:
∫ from 1/3 to 1/2 of 3x^2 dx = [x^3] from 1/3 to 1/2 = (1/2)^3 - (1/3)^3 = 1/8 - 1/27 = 0.04167.
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