PROBABILITY 08
Continuous Distributions
For a continuous random variable, probability is assigned to intervals through a probability density function.
Probability density
A density \(f_X(x)\) satisfies
\[f_X(x)\ge0,\qquad \int_{-\infty}^{\infty}f_X(x)\,dx=1.\]Probability over an interval is
\[\boxed{P(a\le X\le b)=\int_a^b f_X(x)\,dx}\]
1. Density is not point probability
For a continuous random variable,
\[P(X=x)=0.\]The density \(f_X(x)\) can be positive, but probability is obtained from area under the density curve.
2. Cumulative distribution function
\[\boxed{F_X(x)=P(X\le x)=\int_{-\infty}^{x}f_X(u)\,du}.\]Where differentiable,
\[f_X(x)=F_X'(x).\]3. Uniform distribution
If \(X\) is uniform on \([a,b]\),
\[f_X(x)=\frac1{b-a},\qquad a\le x\le b.\]4. Normal distribution
A normal random variable has a bell-shaped density. We write
\[X\sim N(\mu,\sigma^2),\]where \(\mu\) is the mean and \(\sigma^2\) the variance.
5. Exponential distribution
For \(X\sim\operatorname{Exp}(\lambda)\),
\[f_X(x)=\lambda e^{-\lambda x},\qquad x\ge0.\]This distribution is important for waiting times in continuous-time stochastic models.
6. Calculus and probability
\[\boxed{\text{density}\xrightarrow{\text{integration}}\text{probability}}\]