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.
Mode of a continuous random variable from a probability density function
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 \(2/8 = 1/4.\)