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The random variable X has mass function

$$p(x) = \begin{cases} 1/2 &:& x = 0\\ 1/3 &:& x = 2\\ 1/6 &:& x = 3\end{cases}$$

$E(g(X))$ where $g(x) = x^2 + 2x + 1$

My answer includes the following

I change 

     x = 0 into 1 since using the formula above I have 0^2 + 2(0) + 1     
     x = 2 into 9 since 2^2 + 2(2) + 1
     x = 3 into 16 since 3^3 + 2(3) + 1

Then to find the expected mean from the mass function, I use this formula $$E(X) = ยต_X = \sum_x x\, P(X = x)$$

$1 * (1/2) + 9 * (1/3) + 16 * (1/6)$

Got it wrong, what is the correct solution and answer

  • 1
    Your solution looks correct to me โ€“ 2017-02-28
  • 0
    Is the answer 6.167 to 3 decimal places? โ€“ 2017-02-28
  • 1
    Yup should be. Do you have a compelling reason to think it's something else? โ€“ 2017-02-28

1 Answers 1

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To find this expected value, we use the following identity for discrete random variables. Given a random variable $X$ with probability mass function $p(x)$ and function $g(\cdot)$ the expected value of the random variable transformed by the function is given by $$E(g(X))=\sum_{i=-\infty}^{\infty}g(i)p (i).$$ In this case, that yields $$1\cdot \frac{1}{2}+9\cdot \frac{1}{3}+16\cdot \frac{1}{6}=6.1\bar{6}$$ as you correctly computed.