In this section, you will learn how to find the mode of a continuous random variable from a probability density function. Before attempting this section, you should be very good at finding derivatives and performing elementary algebra.

In this section, you will learn how to find the mode of a continuous random variable from a probability density function. Before attempting this section, you should be very good at finding derivatives and performing elementary algebra.

Conclude by giving your students these challenges:

- A Chordingly by NRICH
- Pumpkin Pie Problem by NRICH
- Olympic Magic by NRICH
- Changing Mean

Challenge from Westdene Primary, Brighton by NRICH

Here's the solution: Start by computing the first few powers of \(7\) and \(9,\) until a cycle is found for each:

$$\begin{align} 7^0 = 1 \\ 7^1 = 7 \\ 7^2 = 49 \\ 7^3 = 343 \\ 7^4 = 2401 \\[1em] 9^0 = 1 \\ 9^1 = 9 \\ 9^2 = 81 \end{align}$$Notice the last digit, of a power of 7, could be \(1, 7, 9, or 3,\) each equally likely. Also, notice the last digit, of a power of 9, could be \(1\) or \(9,\) each equally likely. By the multiplication principle, there are thus \(4 \cdot 2 = 8\) pairings, \(2\) of which, have a sum whose last digit is \(0,\) or in other words, is a multiple of \(10.\) Thus, the probability that \(7^x + 9^y\) is a multiple of \(10,\) is \(2/8 = 1/4.\)