Continuous random mode

The mode of a continuous random variable is the value of the variable that has the highest probability density.

We can find the mode by differentiating the probability density function of the variable and then finding the value that gives us the maximum value of the PDF.

The stationary point will be where the derivative is zero - BUT we need to check whether it’s a maximum or minimum - however we want to, for example, by finding the second derivative or just sketching the graph to see.

Sometimes, the mode of a continuous random distribution between two points is at the start or end of the range - that’s our mode, even if the derivative is not zero there.