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Finding the Median, Mean and Mode of a Continuous Distribution

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 solveWeexpand 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 functionissymmetric about

Example: Find the median of the probability distribution

We need to solveWeexpand 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 aboutsothat half the area is on either side.

The mean is given bywhereandarethe 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.