Continuous Probability

probability

Suppose X has a continuous probability density function PDF: f(x)

Area between [x1 and x2] under the curve of f(x) = probability of random variable X.

${\int }_{x_1}^{x_2}f(x)dx=X\in [x1,x2]$

Integral over the entire f(x) should be 1

${\int }_{-\infty }^{+\infty }f(x)dx=1$

Expected Value of g(X):

$\mathbb{E}[g(x)]={\int }_{-\infty }^{+\infty }g(x)f(x)dx$

Mean / Expected Value

$\mu =\mathbb{E}[x]={\int }_{-\infty }^{+\infty }xf(x)dx$

Linearity of Expectation

:LiBatteryWarning: Important Fact: Expected Value is a LINEAR Function and it holds whether the random variables are independent or not.

$\mathbb{E}[aX+bY]=a\mathbb{E}[X]+b\mathbb{E}[X]$

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Variance

A measure of how spread out the data points are in a set. It indicates how far each data point deviates from the mean (expected value). “Describes the dispersion of all values in the dataset.” ${\sigma }^2=\text{Variance}=\mathbb{E}[(X-\mu {)}^2]=\mathbb{E}[X^2]-{\mu }^2$