已知随机变量X的期望与方差，求E((X-a)²)的计算方法

Hey! Let's solve this problem step by step using the hints you provided—it's actually pretty straightforward once you unpack it.

Step 1: Expand the squared term #

First, let's expand the expression inside the expectation:
$$(X - a)^2 = X^2 - 2aX + a^2$$

Step 2: Apply linearity of expectation #

Using the linearity property of expectation ($E(Y_1 + Y_2 + Y_3) = E(Y_1) + E(Y_2) + E(Y_3)$), we can split the expectation into three parts:
$$E((X - a)^2) = E(X^2) - 2aE(X) + E(a^2)$$

Since $a$ is a constant, $E(a^2) = a^2$, and we already know $E(X) = 7$, so this simplifies to:
$$E((X - a)^2) = E(X^2) - 14a + a^2$$

Step 3: Use the variance formula to find $E(X^2)$ #

The variance formula given is:
$$var(X) = E(X^2) - [E(X)]^2$$
We can rearrange this to solve for $E(X^2)$:
$$E(X^2) = var(X) + [E(X)]^2$$

Plugging in the known values ($var(X)=6$, $E(X)=7$):
$$E(X^2) = 6 + 7^2 = 6 + 49 = 55$$

Step 4: Substitute back to get the final result #

Now replace $E(X^2)$ in our earlier expression:
$$E((X - a)^2) = 55 - 14a + a^2$$

We can also write this in standard quadratic form if needed:
$$E((X - a)^2) = a^2 - 14a + 55$$

内容的提问来源于stack exchange，提问作者Eric