Continuous Random Variables

# Continuous Random Variables
### Dr. Dogucu

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<div class="my-footer"> 
 Copyright &copy; <a href="https://mdogucu.ics.uci.edu">Dr. Mine Dogucu</a>. <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a></div>

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## Continuous Random Variables

A continuous random variable `$X$` would have a sample space ( `$S_X$` ) that is uncountably infinite.

Let X be the the proportion of bike owners on campus. Then `$S_x = [0, 1]$`.

Let Y be the the survival time after some surgery. Then `$S_Y = [0, \infty)$`.

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## Probability Density Function (pdf) - `$f(x)$`

A probability __density__ function gives the relative likelihood of the continuous random variable within the sample space.

`$$f(x) \geq0 \text{ for all } x \ \epsilon \ S_X$$`

`$$\int_{x \ \epsilon \ S_X} f(x)dx = 1$$`
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`$$P(X\ \epsilon \ B) = \int_{x \ \epsilon \ B} f(x)dx$$`
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## Example - pdf

<img src="img/cont_f.png" width="90%" style="display: block; margin: auto;" />
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.pull-right[
<img src="slide-3-cont-rv_files/figure-html/unnamed-chunk-2-1.png" style="display: block; margin: auto;" />
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`$x^2 \geq0$`

`$(1-x) \geq0$`

--
<svg style="height:0.8em;top:.04em;position:relative;" viewBox="0 0 512 512"><path d="M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z"/></svg> `$f(x) \geq0 \text{ for all } x \ \epsilon \ S_X$`

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## Area Under the Curve = 1

<img src="img/cont_f.png" width="90%" style="display: block; margin: auto;" />
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.pull-right[
<img src="slide-3-cont-rv_files/figure-html/unnamed-chunk-4-1.png" style="display: block; margin: auto;" />
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`$\int_0^1 12(x^2)(1-x)dx$`
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`$12\int_0^1 (x^2-x^3)dx$` 
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`$12\big[\frac{x^3}{3} -\frac{x^4}{4}\bigg\rvert_0^1\big] = 1$`
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## Probability is Area Under the Curve

<img src="slide-3-cont-rv_files/figure-html/unnamed-chunk-6-1.png" style="display: block; margin: auto;" />
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`$P(0.25<X<0.50) =$`

`$\int_{0.25}^{0.50} 12(x^2)(1-x)dx$`

`$12\int_{0.25}^{0.50} (x^2-x^3)dx$`

`$12\big[\frac{x^3}{3} -\frac{x^4}{4}\bigg\rvert_{0.25}^{0.50}\big] = 0.2617188$`

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## `$P(X=x_i) = 0$`

<img src="slide-3-cont-rv_files/figure-html/unnamed-chunk-8-1.png" style="display: block; margin: auto;" />
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`$P(X=0.40) =$`

`$\int_{0.40}^{0.40} 12(x^2)(1-x)dx$`

`$12\int_{0.40}^{0.40} (x^2-x^3)dx$`

`$12\big[\frac{x^3}{3} -\frac{x^4}{4}\bigg\rvert_{0.40}^{0.40}\big] = 0$`
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## cdf

<img src="slide-3-cont-rv_files/figure-html/unnamed-chunk-10-1.png" style="display: block; margin: auto;" />
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`$P(X\leq x) = \int_l^x f(t)dt$`

`$P(X\leq 0.70) =$`

`$\int_{0}^{0.70} 12(t^2)(1-t)dt$`

`$12\big[\frac{t^3}{3} -\frac{t^4}{4}\bigg\rvert_{0}^{0.70}\big] = 0.6516996$`

Note: `$f(x) = \frac{dF(x)}{dx}$`

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## Expected Value

`$E(X) = \int_{x \ \epsilon \ S_X} xf(x)dx$`

`$\int_0^1 x12(x^2)(1-x)dx$`

`$12\int_0^1 (x^3-x^4)dx$`

`$12\big[\frac{x^4}{4} -\frac{x^5}{5}\bigg\rvert_0^1\big] = 0.6$`

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## Variance

`$Var(X) = E(X^2)- [E(X)]^2$`

`$E(X^2) = ?$`

`$E(X^2) = \int_0^1 x^212(x^2)(1-x)dx$`

`$E(X^2) = 12\int_0^1 (x^4-x^5)dx$`

`$E(X^2) = 12\big[\frac{x^5}{5} -\frac{x^6}{6}\bigg\rvert_0^1\big] = 0.4$`

`$Var(X) = 0.4- 0.6^2 = 0.04$`