<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/59eb822c-9d46-4882-b4ac-33b39cba4e7f/Continuous_random_variables.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/59eb822c-9d46-4882-b4ac-33b39cba4e7f/Continuous_random_variables.png" width="40px" /> Continuous random variable: a random variable that has only continuous values

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• 🦤 The uniform distribution
• 🐣 The Normal Distribution
• 🐔 Exponential Distribution
• 🐟 Poisson distribution
• 🦆 The binomial distribution

The probability density function (PDF)

<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/7c3be66e-600a-4066-802a-69d6337f80e6/PDF.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/7c3be66e-600a-4066-802a-69d6337f80e6/PDF.png" width="40px" /> PDF:

$$ f(x)=\lim_{\delta x\to0}\frac{P(x<X\le x+\delta x)}{\delta x} $$

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• For a continuous random variable

$$ P(a<X<b)=\int^b_af(x)\,\text{d}x $$

• Expectation

$$ \left<X\right>=\int_{-\infin}^{+\infin}xf(x)\,\text{d}x $$

• variance

$$ \begin{aligned} \text{Var}(X)\equiv\sigma^2&=\underbrace{\left<X^2 \right>} -\left< X \right>^2 \\ &\quad\,\left<X^2 \right>=\int_{-\infin}^{+\infin}x^2f(x)\,\text{d}x \end{aligned} $$

• general case

$$ \left<g(X) \right>=\int_{-\infin}^{+\infin}g(x)f(x)\,\text{d}x $$

Cumulative distribution

<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/0cdf3d88-bf21-45af-8a65-8c3b5e5a30c1/CDF.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/0cdf3d88-bf21-45af-8a65-8c3b5e5a30c1/CDF.png" width="40px" /> Cumulative distribution function (CDF):

$$ C(x)=P(X\le x) $$

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• Using

$$ C(b)-C(a)=P(a<X\le b) $$