Question

Question 4:

Let X be a random variable with the following probability distribution:

X -3 6 9

P(X=x) 1/6 1/2 1/3

If the function of random variable is defined as f(X) = (2X+1)2, find µg(X).

Answer 1

To find $\mu_{g(X)}$ where $g(x) = (2x+1)^2$

Here $\mu_{g(X)}$ means expected value of g(x)

$\mu_{g(X)} = \sum g(X) f(X) \\ f(x) = (2x+1)^2 \\ f(-3) = (2(-3) +1)^2 = (-5)^2 = 25 \\ f(6) = (2 \times 6 +1)^2 = 13^2 = 169 \\ f(9) = (2 \times 9 +1)^2 = 19^2= 361 \\ \mu_{g(X)} = 25 \times \frac{1}{6} + 169 \times \frac{1}{2} + 361 \times \frac{1}{3} \\ = 4.16+84.5+120.33 \\ = 208.99 ≈209$