Video Transcript
Let π be a continuous random
variable with the probability density function π 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 π 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 π is
greater than 64. Weβre given a probability density
function, so letβs 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 π 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 β. 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 π 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 thatβs eight over 63 squared
units. Hence, the probability that π 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.
