expectation

introduction

%%visits: 3 This is the weighted average, or the (arithmetic mean?). ## intuition Linearity - 𝔼(aX + bY) = a𝔼(X) + b𝔼(Y) ## rigour If X and Y are joint distributions, then 𝔼(X) = ∫_−∞^∞∫_−∞^∞xf(x, y)dydx = ∫_−∞^∞x∫_−∞^∞f(x, y)dydx = ∫_−∞^∞xf_X(x)dx

𝔼(XY) = ∫_−∞^∞∫_−∞^∞xyf(x, y)dydx ## exam clinic Question 1 from keats quiz - If X is a random variable with probability density function f(x) = 2x, for 0 < x < 1, then what is E(X²) - The formula E(X²) = ∫_−∞^∞x²f(x) gets us all the way to solving. - = $\int_{0}^{1} x^2 2x =\int_{0}^{1} 2x^3 = \frac{1}{2}x^{4}\vline_{x=0}^{x=1}$ - $= \frac{1}{2}$

Let X ≥ 0 be a non-negative integer valued random variable. Prove that $\mathbb{E}[X] = \sum _{k=1}^{\infty} \mathbb{P}(X\ge k)$

$\sum _{k=1}^{\infty} \mathbb{P}(X\ge k = \sum _{k=1}^{\infty} \sum _{j=k}^{\infty} \mathbb{P}(X=j)$ (now we want to swap the order due to foresight)

Now $= \sum _{j=1}^{\infty} \sum _{k=1}^{j} \mathbb{P}(X=j)$

to illustrate this, think of for loops?

k = 1: j = 1,2,3,4,5 .. k = 2: j = 2,3,4,5 … k = 3: j = 3,4,5,6,

…

j = 1 k = 1 j = 2 k = 1,2 … ## examples Example of non-defined expectation $X~f_X(X) = \frac{a}{x^{1+a}},a>0$ an 0 otherwise.

$\mathbb{E}[X] = \int_{x=1}^{\infty}x \frac{a}{x^{1+a}}dx = \int_{x=1}^{\infty}\frac{a}{x^{a}}dx = a \frac{x^{a}}{-a+1}$ or ln (|x|) when a = 1, hence infinity at a=1.

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