The mode of a continuous distribution is the mostprobable value the distribution can take. The probability functionhas a maximum value at this point, so we are finding a stationarypoint for the probability distribution – assuming there is only oneturning point, or at least only one maximum.
The median of a probability distribution is the halfwaypoint. Half the values lie either side of the median.
Example: Find the mode of the probability distribution
We need to solve
Weexpand the brackets to obtain
The mode is the midpoint of the interval [0,1] over which thedistribution is defined. This is to be expected since the function
issymmetric about
Example: Find the median of the probability distribution
We need to solve
Weexpand the brackets to obtain
The expression above factorises to give
The median is the midpoint of the interval [0,1] over which thedistribution is defined. This is to be expected since the functionissymmetric about
sothat half the area is on either side.
The mean is given by
where
and
arethe upper and lower limits of the distribution, the minimum andmaximum values the random variable can take respectively.
In general the lower or upper limits of the above integrals may beinfinity.