Science:Math Exam Resources/Courses/MATH103/April 2017/Question 01 (e) (i)

MATH103 April 2017

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[hide] Question 01 (e) (i)
Plots (A-D) depict probability density functions, pdf ’s, for $-\infty <x<\infty$ at the same scale. List all pdf ’s that satisfy the following criteria: (${\bar {x}}$ denotes the mean and $x_{med}$ the median) (i) ${\bar {x}}=x_{med}$?

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[show] Hint

First we need to clear the two definitions of mean and median for pdf. The mean or average value of $x$ (also called the expected value) in the interval $[a,b]$, denoted ${\bar {x}}$ is given by ${\bar {x}}=\int _{a}^{b}xp(x)dx$ The median is the value of $x$ in the interval $[a,b]$ that splits the probability distribution into two portions whose areas are identical. $\int _{a}^{x_{m}}p(x)dx=\int _{x_{m}}^{b}p(x)dx=0.5$

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[show] Solution

The median simply indicates a place at which the “total mass” is subdivided into two equal portions. (In the case of probability density, each of those portions represents an equal area,$A1=A2=1/2$ since the total area under the graph is 1 by definition.) Simply put, the mean contains more information about the way that the distribution is arranged spatially. This stems from the fact that the mean of the distribution is a “sum” - i.e. integral of terms of the form $xp(x)\Delta x$. Thus the location along the x axis, x, not just the “mass”, $p(x)\Delta x$ , affects the contribution of parts of the distribution to the value of the mean; moving one half of a distribution away from the median will change its mean but not its median. For symmetric distributions, the mean and the median are the same. $A,C$ As for B, the mean will on the left side of median: the point that separates the area into two equal parts will on the left side of y-axis, while the mean will be even further left because it is the integral of terms of the form $xp(x)\Delta x$. It is not only dragged by $p(x)\Delta x$ like the median, but also by $x$. In other words, the ``tail" of the distribution is larger on the left side than on the right side. As for D, like above, just now opposite direction, the mean will be on the right side of the median. answer: $\color {blue}AC$