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The probability density function of a continuous random variable π₯ is as follows: π of π₯ is equal to two π₯ plus one over 18 if π₯ is greater than or equal to one and less than or equal to four and zero otherwise.
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Find the probability that π₯ is less than two and the probability that π₯ is greater than two but less than four.
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For our continuous random variable π₯, the probability that π₯ takes a value in a particular interval can be found by finding the area below the graph of its probability density function π of π₯ over that range of π₯-values.
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In this question, the probability density function is a linear function of π₯.
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And so, its graph will be a straight line.
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The probability that π₯ is less than two first of all then will be the area under this graph between the values of one and two.
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And we can see that this shape will be a trapezium.
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So we can work out its area using our formulae for the areas of 2D shapes.
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In general, the area of a trapezium is found by finding half of the sum of the parallel sides which are often labelled as π and π and multiplying this by the perpendicular distance between them, the height of the trapezium.
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In our question, the height of this trapezium is the difference between the values of one and two.
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So β is equal to one.
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The values of π and π, the lengths of the two vertical sides of this trapezium, will be the values of the function π of π₯ evaluated at one and two, respectively, π of one and π of two.
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To find each of these values, we can substitute into the probability density function.
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π of one is equal to two multiplied by one plus one over 18 which simplifies to three over 18.
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In the same way, π of two is equal to two multiplied by two plus one over 18 which simplifies to five over 18.
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So we can now substitute into our formula for the area of a trapezium.
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The area is equal to one-half multiplied by three 18ths plus five 18ths multiplied by one.
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Three 18ths plus five 18ths is equal to eight 18ths.
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And we can then cancel a factor of two from the two in the denominator and the eight in the numerator, giving four 18ths.
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We can cancel a further factor of two to give two in the numerator and nine in the denominator.
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The area is therefore equal to two-ninths.
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Remember this area gives the probability that our continuous random variable π₯ is less than two.
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And so, we found that the probability that π₯ is less than two is equal to two-ninths.
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For the second part of the question, we are asked to find the probability that π₯ is greater than two but less than four.
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So weβre looking for the area under this straight line between π₯-values of two and four.
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Thatβs the area that Iβve now shaded in pink.
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We could apply the same method that we just used to find the probability that π₯ is less than two by finding the area of this trapezium.
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But there is in fact an easier way.
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For any continuous random variable, the sum of all probabilities must be equal to one.
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And therefore, the full area below the graph of its probability density function must also be equal to one.
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We can therefore find the probability that π₯ is greater than two, but less than four by subtracting the probability weβve just found for the probability that π₯ is less than two from one.
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This gives one minus two-ninths which is equal to seven-ninths.
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You can of course confirm this by calculating the area of this trapezium if you wish.
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Now, there is actually a more general method that we can use to find these areas.
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And it would certainly be essential to use this method if the graph of the probability density function was not a straight line.
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To find the area below a curve, we can use integration so the probability that π₯ is greater than two but less than four could be found by integrating the probability density function π of π₯ with respect to π₯ between the limits of two and four.
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In the same way, the probability that π₯ is less than two could be found by integrating the probability density function with respect to π₯ between the limits of one and two.
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If you are familiar with integration, then you could perform this integration and confirm that it does indeed give the same answers.
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We found using our method of the area of a trapezium that the probability π₯ is less than two is two-ninths and the probability that π₯ is greater than two but less than four is seven-ninths.