Finding the mode from a p.d.f.

Identifying the Problem Type

Look carefully at the problem description. If it mentions a probability density function (p.d.f.), and asks you to find the mode, then you’re dealing with a task of finding the mode from a p.d.f.
Be on the lookout for other relevant terms: “density function”, “maximum likelihood”, “probability function”, or “highest probability”.

Begin by understanding the density function provided. Remember, the mode of a continuous probability density function is the highest point on the graph, where the function attains its maximum value.
Pay close attention to the variables. Which one represents the variable for which you need to find the mode? What is the dependent variable that represents the probability density for each value of the independent variable?

To find the mode, you should set the derivative of the p.d.f. equal to zero and solve for the variable. This gives the point(s) where the p.d.f. reaches its maximum.
Careful calculations are essential. Make sure you follow all the necessary steps and keep track of all algebraic manipulations accurately to lead to the correct solution.

When you find the point(s) that satisfy the condition for the derivative equaling zero, these point(s) are either maximum, minima, or points of inflection. To confirm this, you might need to use the second derivative test.
Always link your results back to the problem. If you found multiple points, consider the relevance to the problem. In many practical scenarios, there would be a single mode, but there can be cases of bimodal or multimodal densities too.

Don’t forget to review your work. Especially for derivative calculations, there is room for errors. A small mistake in calculus can lead to the wrong inference about the mode, making your analysis incorrect.
After calculating the mode, make sure it’s a point where the p.d.f. indeed has its highest value. You can confirm this by checking the p.d.f.’s value at the calculated mode as well as values at nearby points.

Lastly, bear in mind that the mode represents the most likely outcome in a probability distribution, but it doesn’t tell the entire story. The spread and shape of the distribution, expressed through other measures like variance and skewness, are also crucial in understanding the nature of the probability event in question. Remember that the mode is only one part of the picture.