Ch6: Continuous Random Variables

• For continous random variables we generally have $P(x)=0$
• For continous variables we need another kind of model, where we find probabilities for x being in an interval. this is a probability density function or pdf

probability density function

non-negative function f, such that $P(a\le x\le b)={\int }_a^{b}f(x)dx$

• In this definition probabilities are areas: the probability that x takes the values between a and b is equal to the area below the graph.
• Open or Close interval does not affect the probability.

Properties

• $f(x)\ge 0$
• ${\int }_{-\mathrm{\infty }}^{+infty}f(x)dx=1$

If a function has these two conditions then it is a pdf

$$E(X)={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}x\cdot f(x)dx$$$$var(X)=E(X-\mu {)}^2$$$$E(g(X))=\int g(x)f(x)dx$$

Distribution Function

Distribution Function

The function F defined by $F(x)=P(X\le x)$ with $x\in \mathbb{R}$, is the cumulative distribution function cdf of the random variable X

Properties

For any distribution function:

• F is non decreasing.
  • if $x_2>x_1,thenF(x_2)\ge F(x_1)$
• $li{m}_{x\to \mathrm{\infty }}F(x)=1$
• $li{m}_{x\to -\mathrm{\infty }}F(x)=0$
• F is continuous from the right
  • $li{m}_{h\to 0+}F(x+h)=F(x)$

Properties

• $P(a\text{l}X\le b)=F(b)-F(a)$
• $P(X>x)=1-F(x)$
• $P(X<x)=li{m}_{h\to 0^{+}}F(x-h)=F(x)$
• $P(X=x)=F(x)-P(X<x)$

INFO

A random variable X is continuous if the distribution function F of X is a continous function.

Properties of Continous distributions

• $P(X=x)=0$
• $P(X\in [a,b])={\int }_a^{b}f(x)dx=F(b)-F(a)$ closed interval equals open interval
• $F(x)={\int }_{-\mathrm{\infty }}^{x}f(u)du$
• $f(x)=\frac{d}{dx}F(x)$
• If the density function f(x) of X is symetric about x = c, then E(X) = c.

Uniform distribution