12Cmesuresofcentreandspread


 * Measures of central tendency and spread **

As with discrete probability distributions we can find measures of central tendency and spread for continuous probability distributions.


 * Measures of central tendency **

__** Mean **__ ... ... The mean (or expected value) of a continuous random variable X, with a probability distribution function f(x), can be found by:

math . \qquad \mu=E(X) = \displaystyle{ \int\limits_{-\infty}^{\infty} \; x \,f(x) \; dx } \qquad. math

... ... {Compare this to the rule for the Expected Value of a discrete random variable}

__** Median **__ ... ... The median, //m//, of a probability distribution function is the value of X such that:

math \\ . \qquad Pr(X \leqslant m) = Pr(X \geqslant m) = 0.5 \qquad. \\ . \\ . \qquad \displaystyle{ \int\limits_{-\infty}^{m} \; f(x) \; dx} = 0.5 math

__** Mode **__ ... ... If //f(x)// is the probability density function of X, then the mode, M, is the value of X such that

math . \qquad f( M ) \geqslant f( x ) \qquad\text{for all other x} \qquad. math

... ... {In other words it is value of x that gives the maximum point on the graph (either a stationary point or an endpoint)}


 * Measures of spread **

__** Interquartile Range **__ ... ... ... The InterQuartile Range is given by Q 3 – Q 1 ,
 * where Q 1 is 1/4 of the way through the data
 * and Q 3 is 3/4 of the way through the data

.. ... ... To find a quartile, we find where the area under the curve is equal to 25% or 0.25

math . \qquad \text{So } \; Q_1 = a \; \text{ where } \; \displaystyle{ \int \limits_{-\infty}^a \; f(x) \; dx} = 0.25 \qquad. \\ . \\ . \qquad \text{And } \; Q_3 = b \; \text{ where } \; \displaystyle{ \int \limits_b^{\infty} \; f(x) \; dx} = 0.25 math

math . \qquad \text{Then } \textbf{ IQR } = b - a \qquad. math

__** Variance **__ ... ... The variance of a continuous random variable X, with a probability distribution function f(x), can be found by:

math . \qquad Var (X) = E\left[(X - \mu)^2\right] \qquad. math . math . \qquad \qquad= \displaystyle{ \int\limits_{-\infty}^{\infty} \; (x - \mu)^2f(x) \; dx} \qquad. math


 * OR**

math . \qquad Var (X) = E \Big( X^2 \Big) - \mu^2 \qquad. math

math . \qquad \qquad= \displaystyle{ \int\limits_{-\infty}^{\infty} \; x^2f(x) \; dx} - {\mu}^2 \qquad. math

... ... {This is also comparable to the rule for Variance of a discrete random variable}

__** Standard deviation **__

math . \qquad \sigma = \sqrt{Var (X)} \qquad. math


 * Example 1 **

The continuous random variable //X// has probability density function given by: math . \qquad f(x)=\Bigg\{ \begin{matrix} \dfrac{k}{x}&1\leqslant x \leqslant 9 \qquad .\\ 0&\text{elsewhere} \qquad. \end{matrix} math

... ... ** (a) ** .. Find the value of k. ... ... ** (b) ** .. Find the mean and variance of X.

__Solution:__

... Since f is a probability density function we expect math . \qquad \displaystyle{ \int\limits_{-\infty}^{\infty} \; f(x) \; dx} =1 \qquad. math so math \\ .\qquad \displaystyle{ \int\limits_{1}^{9} \; \dfrac{k}{x} \; dx} =1 \qquad. \\ . \\ . \qquad \Big[ k \log_e(x) \Big]_1^9 = 1 \\. \\ . \qquad k\log_e(9)=1 \\. \\ . \qquad k=\dfrac{1}{\log_e(9)} \\. \\ . \qquad k = \dfrac{1}{\log_e \big( 3^2 \big) } \\. \\ . \qquad k = \dfrac{1}{2 \log_e(3)} math
 * (a) **


 * (b) **

math \\ \mu = \displaystyle{ \int\limits_{1}^{9} \; xf(x)\; dx} \\. \\ .\quad= \displaystyle{ \int\limits_{1}^{9} \; x\times\dfrac{1}{2x \log_e(3)} \; dx} \qquad. \\ . \\ .\quad= \displaystyle{ \int\limits_{1}^{9} \; \dfrac{1}{2 \log_e(3)} \; dx} \\. \\ .\quad= \left[\dfrac{x}{2 \log_e(3)}\right]_{1}^{9} math

math .\quad= \dfrac{9}{2 \log_e(3)}-\dfrac{1}{2 \log_e(3)} \qquad \qquad. \\ . \\ .\quad= \dfrac{8}{2 \log_e(3)} \\. \\ . \quad = \dfrac{4}{\log_e(3)} math

math \\ Var (X) = E(X^2) - {\mu}^2 \\. \\ .\qquad= \displaystyle{ \int\limits_{1}^{9} \; x^2f(x) \; dx} - {\mu}^2 \\. \\ .\qquad= \displaystyle{ \int\limits_{1}^{9} \; x^2 \times \dfrac{1}{2x \log_e(3)} \; dx} - \left( \dfrac{4}{\log_e(3)} \right)^2 \qquad. \\ . \\ .\qquad= \displaystyle{ \int\limits_{1}^{9} \; \dfrac{x}{2\log_e (3)} \; dx} - \left( \dfrac{4}{\log_e(3)} \right)^2 \\. \\ .\qquad= \left[ \dfrac{x^2}{4\log_e(3)} \right]_{1}^{9} - \left( \dfrac{4}{\log_e(3)} \right)^2 math

math .\qquad= \dfrac{81}{4 \log_e(3)} - \dfrac{1}{4 \log_e(3)} - \left( \dfrac{4}{\log_e(3)} \right)^2 \qquad. \\ . \\ .\qquad= \dfrac{80}{4 \log_e(3)} - \left( \dfrac{4}{\log_e(3)} \right)^2 \\. \\ .\qquad= \dfrac{20}{\log_e(3)} - \left( \dfrac{4}{\log_e(3)} \right)^2 math

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