Video: MATH-STATS-2018-S1-Q13

MATH-STATS-2018-S1-Q13

05:13

Video Transcript

Suppose that π‘₯ is a continuous random variable with probability density function 𝑓 of π‘₯ equals π‘₯ plus one over 12. If π‘₯ is greater than or equal to zero and less than or equal to four and zero otherwise. Find the probability that π‘₯ is less than two and the probability that π‘₯ is greater than two but less than five.

For a continuous random variable, probabilities can be found by finding the area below the graph of their probability density function. For this continuous random variable, the probability density function 𝑓 of π‘₯ is π‘₯ plus one over 12 if π‘₯ is between zero and four inclusive, which is a straight line. And the probability density function is zero otherwise.

To find the probability that π‘₯ is less than two, we need to find the area below the graph between zero and two. That’s the area that I’ve shaded in orange. Now taking a closer look at this area, we can see that it is a trapezium. And we have a formula for calculating the area of a trapezium. It’s a half multiplied by π‘Ž plus 𝑏 multiplied by β„Ž, where π‘Ž and 𝑏 are the two parallel sides and β„Ž is the distance between them.

In our trapezium, the value of β„Ž is two because it’s the difference between zero and two. The values of π‘Ž and 𝑏 will be the heights of these two vertical lines, which will be 𝑓 of zero and 𝑓 of two, respectively. We can find 𝑓 of zero and 𝑓 of two by substituting into the probability density function. 𝑓 of zero is equal to zero plus one over 12, which is just equal to one twlevth. 𝑓 of two is equal to two plus one over 12, which is equal to three twelvths. Now we could simplify this fraction to one-quarter, but we’ll keep it with a denominator of 12 for now because then it’s consistent with the denominator of 12 that we have for 𝑓 of zero.

So, now that we know all three of the measurements for our trapezium, we can work out its area. We have a half multiplied by one-twelvth plus three-twelvth multiplied by two. The half and the two will cancel each other out. So, we’re just left with one twelvth plus three twelvths. This is equal to four twelvths, which can be simplified to one-third by dividing both the numerator and denominator of the fraction by four.

So, this gives us our answer to the first part of the question. The area below the graph between zero and two is one-third. So, the probability that our continuous random variable π‘₯ is less than two is equal to one-third.

Next, we’re asked to find the probability that π‘₯ is greater than two but less than five. Now notice that our probability density function is zero for all values of π‘₯ which are greater than four, which means there is no area between four and five. So, the probability that π‘₯ is greater than two but less than five is actually equal to the probability that π‘₯ is greater than two but less than or equal to four, which is given by the area between two and four on our graph of the probability density function. That’s the area that I have shaded in pink.

Now this area is again a trapezium, so we could work it out using the same method that we did for the first part of the question. But an easier method is to remember that the sum of all probabilities in a probability density function must be equal to one. And therefore, the area below the full graph of a probability density function is also equal to one.

So, the orange and the pink areas must both sum to one, which means we can work out the probability that π‘₯ is greater than two but less than five by subtracting the area we’ve just found, that’s one-third, from one. One minus a third is two-thirds. So, we’ve found that the probability that π‘₯ is greater than two and less than five, which is also the probability that π‘₯ is greater than two and less than or equal to four, is equal to two-thirds.

Now there is actually one other method that we could’ve used to answer this question. And it’s a more general method for finding probabilities for a continuous random variable. Suppose that the probability density function was a curve rather than a straight line. Well, we can find the area below a curve using integration. The probability that π‘₯ is greater than some value π‘Ž and less than some value 𝑏 can be found by integrating the probability density function 𝑓 of π‘₯ between π‘Ž and 𝑏. So, for example, the probability that π‘₯ is less than two could be found by evaluating the definite integral of π‘₯ plus one over 12 between zero and two.

If you’re familiar with integration, then you could confirm that this does indeed give the same answer as we found by calculating the area of the trapezium. Integration just gives a more general method for finding probabilities for a continuous random variable. But it wasn’t necessary in this case. We found that the probability that π‘₯ is less than two is one-third and the probability that π‘₯ is greater than two but less than five is two-thirds.

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