Lecture 7 Quantile/2017 2018 /

Definition (( Quantile )) The qth quantile of a random variable X or of
its corresponding distribution is denoted by eq and is defined as the smallest number es atisfying Fx(e) > q.
If X is a continuous random variable, then the qth quantile of X is given as the smallest number satisfying Fx(e) = q. Definition 18 Median The median of a random variable X, denoted by
medx, med (X), or (so, is the .5th quantile. Remark In some texts the median of X is alternatively defined as any number, say med (X), satisfying P[X ~ med (X)] ?1/2 and P[X > med (X)] ? 1/2 •
So the median of X is any number that has half the mass of X to its right and the other half to its left, which justifies use of the word H median." We have already mentioned that8[X], the first moment, locates the" center" of the density of X. The median of X is also used to indicate a central location of the density of X. A third measure of location of the density of X, though not necessarily a measure of central location, is the mode of X, which is defined as that point (if such a point exists) at whkh fx( ) attains its maximum.
Other measures of location [for example, t( .25 + .75)] could be devised, but three, mean, median, and mode, are the ones commonly used. We previously mentioned that the second moment about the mean, the variance of a distribution, measures the spread or dispersion of a distribution. Let us look a little further into the manner in which the variance characterizes the distribution.