A Level Maths Notes: S2 – Finding the Median, Mode and Mean of a Continuous Distribution
The mode of a continuous distribution is the most probable value the distribution can take. The probability function has a maximum value at this point, so we are finding a stationary point for the probability distribution – assuming there is only one turning point, or at least only one maximum.
The median of a probability distribution is the halfway point. Half the values lie either side of the median.
Example: Find the mode of the probability distribution
We need to solve
We
expand the brackets to obtain
The mode is the midpoint of the interval [0,1] over which the
distribution is defined. This is to be expected since the function
is
symmetric about
Example: Find the median of the probability distribution
We need to solve
We
expand the brackets to obtain
The expression above factorises to give
The median is the midpoint of the interval [0,1] over which the
distribution is defined. This is to be expected since the functionis
symmetric about
so
that half the area is on either side.
The mean is given by
where
and
are
the upper and lower limits of the distribution, the minimum and
maximum values the random variable can take respectively.
In general the lower or upper limits of the above integrals may be infinity.