Video Transcript
Let 𝑋 be a continuous random
variable with the probability density function 𝑓 of 𝑥 is equal to one over 63 when
𝑥 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 𝑥 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 𝑥 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 𝑥 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.
