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​​Discussion 11B

​​Tail Sum Formula

​​Discrete

​​If ​​$X$​ only takes on values in the natural numbers then

​​​​$E[X] = \sum_{k=1}^\infty x \Pr(X = k) = \sum_{k=1}^\infty \Pr(X \geq k)$​

​​Continuous

​​If ​​$X$​ only takes on nonnegative values then

​​​​$E[X] = \int_0^\infty x f_X(x) dx = \int_0^\infty \Pr(X \geq x) dx = \int_0^\infty (1 - F_X(x)) dx$​

​​Joint Continuous Distributions

​​​​$f_{X,Y}(x,y)$​ is the PDF of the joint distribution of ​​$X$​ and ​​$Y$​ where ​​$\Pr(X \in [a, b], Y\in [c, d]) = \int_c^d \int_a^b f_{X,Y}(x,y) dx dy$​.

​​​​$f_{X,Y}(x,y) \geq 0$​

​​​​$\int_0^\infty \int_0^\infty f_{X,Y}(x,y) dx dy = 1$​

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​​​​$X$​ and ​​$Y$​ are independent when ​​$f_{X,Y}(x,y) = f_X(x) f_Y(y)$​.

​​Finding Distributions

​​To find the distribution of a continuous random variable ​​$X$​, find the CDF ​​$F(x) = \Pr(X < x)$​, then differentiate to get the PDF ​​$f(x)$​.

​​Normal Distribution

​​​​$X \sim \mathcal{N}(\mu, \sigma)$​

​​​​$f(x) = \frac{1}{\sqrt{2 \pi \sigma^2}} \exp{\frac{-(x-\mu)^2}{2 \sigma^2}}$​

​​​​$X$​ has mean ​​$\mu$​ and variance ​​$\sigma^2$​.

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​​​​$Z \sim \mathcal{N}(0,1)$​ has the standard normal distribution.

​​If ​​$X = a Z + b$​ then ​​$X \sim \mathcal{N} (b, a)$​.

​​If ​​$X\sim \mathcal{N}(\mu_X, \sigma_X)$​ and ​​$Y \sim \mathcal{N} (\mu_Y, \sigma_Y)$​ are independent then ​​$X + Y \sim \mathcal{N} (\mu_X + \mu_Y, \sqrt{\sigma_X^2 + \sigma_Y^2})$​.

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