Step By Step Calculus » 19.4 - Probability

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Synopsis
A continuous random variableXX is a variable that can take any value from the set of real numbers. For a continuous variable, there must exist a probability density functionf(x)f(x) which is nonnegative and for which \displaystyle \int_{-\infty}^{\infty} f(x)dx =1 .\displaystyle \int_{-\infty}^{\infty} f(x)dx =1 . The probability that XX will take a value from interval I=[a,b]I=[a,b] is given by
\displaystyle P(a\leq X\leq b)=\int_a^b f(x) dx.
\displaystyle P(a\leq X\leq b)=\int_a^b f(x) dx.
The expected value or average or mean of a continuous random variable XX is given by
\displaystyle E[X]=\int_{-\infty}^{\infty} xf(x) dx.
\displaystyle E[X]=\int_{-\infty}^{\infty} xf(x) dx.
If mm is the median of a continuous random variable XX, then \displaystyle \int_m^{\infty} f(x)dx=\frac{1}{2}.\displaystyle \int_m^{\infty} f(x)dx=\frac{1}{2}.
A continuous random variable XX is said to be exponentially distributed if its probability density function is given by
\displaystyle f(x)=\left\{\begin{array}{ll} \frac{1}{\mu}e^{-x/\mu} & x\geq 0 \\ 0 & x <0 \end{array} \right.
\displaystyle f(x)=\left\{\begin{array}{ll} \frac{1}{\mu}e^{-x/\mu} & x\geq 0 \\ 0 & x <0 \end{array} \right.
where \mu\mu is the mean value of X.X.
A continuous random variable XX is said to be uniformly distributed over the interval (a,b)(a,b) if its probability density function is given by
\displaystyle f(x)=\left\{\begin{array}{ll} \frac{1}{b-a} & a<x<b \\ 0 & \textrm{otherwise} \end{array} \right. .
\displaystyle f(x)=\left\{\begin{array}{ll} \frac{1}{b-a} & a<x<b \\ 0 & \textrm{otherwise} \end{array} \right. .
A continuous random variable XX is said to be normally distributed if its probability density function is given by
\displaystyle f(x)=\frac{1}{\sqrt{2\pi}\;\sigma}e^{-(x-\mu)^2/\sigma}\qquad -\infty<x<\infty
\displaystyle f(x)=\frac{1}{\sqrt{2\pi}\;\sigma}e^{-(x-\mu)^2/\sigma}\qquad -\infty<x<\infty
where \mu\mu and \sigma\sigma are the mean and standard deviation of XX respectively.