Question Video: Using Probability Density Function of Continuous Random Variable to Find Probabilities Mathematics

Let 𝑋 be a continuous random variable with the probability density function 𝑓(π‘₯) = 1/63, 9 ≀ π‘₯ ≀ 72 and 𝑓(π‘₯) = 0, otherwise. Find 𝑃(𝑋 > 64).

01:49

Video Transcript

Let 𝑋 be a continuous random variable with the probability density function 𝑓 of π‘₯ is equal to one over 63 when π‘₯ is greater than or equal to nine and less than or equal to 72 and zero otherwise. Find the probability that 𝑋 is greater than 64.

In this example, we need to find the probability of an event for a continuous random variable when the event is 𝑋 is greater than 64. We’re given a probability density function, so let’s begin by graphing this function. The function takes the value one over 63 when π‘₯ is between nine and 72 and a zero otherwise.

We recall that the probability of an event for a continuous random variable is given by the area under the graph of the probability density function 𝑓 of π‘₯ over the interval representing the event. In our case then, we need to find the area under this graph over the interval 64 to ∞. However, since we know that this function is equal to zero for π‘₯ greater than 72, we need only find the area under the curve for 𝑋 between 64 and 72. That is the area of the highlighted area on the graph, which is a rectangle. And this area gives us the probability of the given event.

We see that the base of the rectangle has length 72 minus 64; that is eight units. And the height of the rectangle is one over 63. And we know, of course, that the area of a rectangle is the base times the height, which in our case is eight multiplied by one over 63. And that’s eight over 63 squared units. Hence, the probability that 𝑋 is greater than 64 is eight over 63. And we note that this is a reasonable answer for a probability, since eight over 63 lies between zero and one.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.