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

# Question Video: Using Probability Density Function of Continuous Random Variable to Find Probabilities Mathematics • Third Year of Secondary School

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Let π be a continuous random variable with the probability density function π(π₯) = 1/63 when 9 β€ π₯ β€ 72 and π(π₯) = 0 otherwise. Find π(π > 64).

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### 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. Weβre given a probability density function, so letβ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 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 thatβs 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.

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