2.2 Review of Continuous random variables

In contrast to discrete random variables, continuous random variables have an infinite set of outcomes, thus each outcome has a probability of 0 of occurring. Instead we have a density to describe the probabilities of ranges of outcomes. The area under this curve over all outcomes is 1 (just as the sum over all outcomes of a discrete random variables is 1).

Some important quantities of continuous random variables are:

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  Expected value: $𝔼\left(X\right)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}xf(x)\text{d}x$.
  Sample mean: $\overline{x}=\frac{1}{n}{\sum }_i{x}_i$.

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  Expected value of a function: $𝔼\left(g(X)\right)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}g(x)f(x)\text{d}x$.

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  Variance $\text{Var}(R)=𝔼\left(R^2\right)-{[𝔼\left(R\right)]}^2$.
  Sample variance: $s_x^2=\frac{1}{n-1}{\sum }_{i=1}^n{(x_i-\overline{x})}^2$.

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  Standard deviation $=\sqrt{\text{Variance}}$.
  Sample standard deviation: $s_x=\sqrt{s_x^2}$.

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  Cumulative Distribution Function (CDF): $F(x)=\mathbb{P}(X\le x)$. Again the CDF can only take values between 0 and 1 and is an increasing function (by the axioms of probability).

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  Probability Density Function (PDF): $f(x)=\frac{\text{d}F(x)}{\text{d}x}$.

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  Using the above the CDF can also be written as $F(x)={\int }_{-\mathrm{\infty }}^{x}f(u)\text{d}u$.

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  Furthermore, $\mathbb{P}(a\le X\le b)={\int }_a^{b}f(x)\text{d}x=F(b)-F(a)$.

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  You should be able to remember and use formulae such as the following. For rvs, X and Y

  $$𝔼\left(aX+bY+c\right)=a𝔼\left(X\right)+b𝔼\left(Y\right)+c\mathit{\hspace{1em}\hspace{1em}}\text{Var}(aX+bY+c)=a^2\text{Var}X+b^2\text{Var}Y+ab\text{Cov}(X,Y).$$

  If the variables are independent (i.e. $\text{Cov}(X,Y)=0$), we also have

  $$\text{Var}(aX+bY+c)=a^2\text{Var}X+b^2\text{Var}Y.$$