 This is a video about the mode of a continuously distributed random variable. Suppose that you're looking at the graph of the probability density function. The mode is the place with the highest probability density. It's the x-coordinate of the highest point on the graph. So let's look at some examples. First of all, what do you think is the mode in this case? This is the graph of the probability density function. Well, the mode is 1 because that's the place with the highest probability density. It's the x-coordinate of the highest point on the graph. Here's another example. What do you think that the mode is in this case? Well, this case is slightly unusual because there are two places with equal highest probability density. Both 0 and 4 have the highest probability density. So we say that the distribution is bimodal and that the two modes are 0 and 4. My next example would be a little bit harder. In this situation, we can't see the mode straight away. We need to know the definition of the probability density function. Let's look at the probability density function and try to work out the mode. f of x is 3 over 64 times x squared times 4 minus x when x is between 0 and 4 and it's 0 otherwise. We need to find the maximum point on this graph. In order to find the maximum, we need to do some differentiating. We need to find the derivative of the probability density function. We need to differentiate 3 over 64 times x squared times 4 minus x. We can take the 3 over 64 outside and say that it's 3 over 64 times the derivative of x squared times 4 minus x which is 3 over 64 times d by dx of 4x squared minus x cubed and that's 3 over 64 times 8x minus 3x squared. So f dash of x is 3 over 64 times 8x minus 3x squared and we know that at the maximum f dash of x will be equal to 0. So we need to solve 8x minus 3x squared equals 0 and we can factorise it to get x times 8 minus 3x equals 0 the solutions of which are when x is equal to 0 and when 3x is equal to 8 or in other words when x is equal to 2 and 2 thirds and clearly this is the x coordinate of the maximum point on the graph so the mode here is 2 and 2 thirds. Let's look at one more example. This time suppose we've got a probability density function which is defined in 3 pieces when x is between 0 and 3 f of x is 2 over 25 times x times 4 minus x and when x is between 3 and 4 f of x is 2 over 25 times x and f of x is 0 everywhere else. In this case it's a bit harder to find the mode because we don't know what the graph looks like. So let's begin by sketching a graph and seeing if that helps. Now for the first bit if we draw y equals 2 over 25 times x times 4 minus x it looks like this. It's a quadratic curve with zeros at x equals 0 and x is equal to 4 and it's symmetric so it clearly has a maximum when x is equal to 2. If we draw y equals 2 over 25 times x that's the second part of the definition that's a straight line which passes through the origin and has gradient 2 over 25. Combining those and remembering that f of x is equal to 0 outside the interval 0 up to 4 this is what the probability density function actually looks like and now we can see that there are two candidates for the mode. It could be that the mode is at 2 but it could also be that the mode is at 4 and we'll have to do some calculation to find out at which point the probability density is actually higher. So we need to remember that the probability density function is given by this definition and we'll work out the probability density at x is equal to 2 f of 2 is equal to 2 over 25 times 2 times 4 minus 2 which is 8 over 25 and we'll also work out the probability density when x is equal to 4 by looking at f of 4 which is 2 over 25 times 4 which is also 8 over 25. So this is another bimodal distribution where the modes are 2 and 4. We could also have done this question without sketching the curve but if you want to do it without sketching the curve you'll need to remember that the maximum point, the highest point on the graph could either be a local maxima where f dashed of x is equal to 0 or it could be at a place that's the end point of one of the intervals in the definition. So what you need to do is to find where f dashed of x is equal to 0 look at the end points of each interval in the definition of the function and then find the probability density at each of these places in order to see where it's highest. OK, this is the end of my video about finding the mode of a continuously distributed random variable. I hope that you found it useful. Thank you very much for watching.