Also called, probability density functions (pdf), densities

• have a continuous

Probabilities can be thought of as areas under the curves of functions (found by integration)

• A single point on the line has no width ($P(X=a)=0$)
• implies $P(X\le a)=P(X<a),P(a\le X\le b)=P(a\le X<b)=P(a<X\le b)=P(a<X<b)$

A function $f(x)$ is called a probability density function (pdf) of the continuous random variable $X$ if and only if

for any real constants $a$ and $b$ with $a\le b$

A function can serve as a probability density of a continuous random variable $X$ if all $f(x)$:

1. $f(x)\ge 0,$ for $-\mathrm{\infty }<x<\mathrm{\infty }$ (pdf will never go below the x axis)
2. ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)dx=1$

• often represented by a probability density function (PDF), f(x) that gives the relative likelihood of all values of x
  • f(x) = height of the curve at point x
    • not a probability
      • probabilities = areas under the curve
  • theoretical mean = E(X)
    • $E(X)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}xf(x)dx$
  • variance = $\text{sigma { <a class="tag" onclick="toggleTagSearch(this)" data-content="\#2">\#2</a>}}$
    • $Var(X)=E[(X-\mu {)}^2]$
      • = ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(x-\mu {)}^{2}f(x)dx$
    • handy relationship
      • $E[(X-\mu {)}^2]=E(X^2)-(E(X){)}^2$
        • $E(x)=\mu$
        • $\text{E[(x-mu)^2] = E[x^2-2xmu+mu { <a class="tag" onclick="toggleTagSearch(this)" data-content="\#2">\#2</a>} ]=E(x^2)-2E(xmu)+E(mu^2)}$
        • $=E(x^2)-2E(x)\cdot \mu +{\mu }^2$
        • $=E(x^2)-2\mu \mu +{\mu }^2=E(x^2)-{\mu }^2=E(x^2)-[E(x){]}^2$

Cumulative Distribution Function (CDF)

For a continuous random variable $X$ that has the probability density at $t$ as $f(t)$, the cumulative distribution of $X$ is

If $f(x)$ and $F(x)$ are the pdf and cdf of $X$, then