Question Video: Finding a Probability for a Continuous Random Variable Mathematics

Let 𝑋 be a continuous random variable with the probability density function 𝑓(π‘₯), represented by the following graph. Find the value of π‘Ž that makes 𝑃(5 < 𝑋 < π‘Ž) = 1/3.

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Video Transcript

Let 𝑋 be a continuous random variable with the probability density function 𝑓 of π‘₯ represented by the following graph. Find the value of π‘Ž that makes the probability that 𝑋 is greater than five but less than π‘Ž equal to one-third.

We recall first that for a continuous random variable 𝑋, the probability that π‘₯ lies in a given interval is equivalent to the area under the graph of its probability density function 𝑓 of π‘₯ between the endpoints of that interval. We’re looking to find the probability that 𝑋 is greater than five and less than some value π‘Ž. So this will correspond to the orange area shown.

We know what we want the area to be. It’s one-third. We also know the width of this rectangle. From the graph of the probability density function, we can see that the height is one-sixth. The base or length of the rectangle is from five to the unknown value π‘Ž. So an expression for the rectangle’s length is π‘Ž minus five. Recalling that the area of a rectangle is found by multiplying its length and width together, we can therefore form an equation. One-sixth multiplied by π‘Ž minus five is equal to one-third. We can then solve this equation to determine the value of π‘Ž.

First, we multiply both sides of the equation by six. On the left-hand side, we have π‘Ž minus five, and on the right-hand side, six multiplied by a third or six over three, which is equal to two. We then add five to each side of the equation, giving π‘Ž equals seven.

So by recalling that for a continuous random variable 𝑋, the probability that 𝑋 lies in a given interval is equal to the area under the graph of its probability density function between the endpoints of that interval, we found the value of π‘Ž such that the probability 𝑋 is greater than five but less than π‘Ž is one-third is seven.

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