Statistics with Probability

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    Ezra Halleck

    2. The probability density function is in the form of an isosceles triangle whose x-intercepts are 3 and 15. Find P(X>7).




    The base will be 12 since 15-3 is the base.
    we know the area is 1 and the formula to find the area of a triangle is 1/2*base*height=Area
    I found the height to be 1/6.
    To find the height for x=7 we can use the similar triangle method.
    8 is the center where the height is 1/6.

    I got the height for x=7 to be 7/48

    P(X>7) will be a bit more than 50% because if we look at it this way: The distribution is made up of two right triangles with the base 6. P(X>7) fills up an entire right triangle. Each of the right triangles have an area .5 because the area of the whole isosceles triangle is 1.

    So we need to find out the remaining part which appears to be a trapezoid.
    Area of a trapezoid can be found by, 1/2*h*(b1+b2).
    To avoid confusion, we need to rotate the trapezoid by 90 degrees.

    The height is 1 because 8-7 is 1
    b1 is 1/6 and b2 is 7/48

    1/2*1*(1/6+7/48)= 0.15625 (I got lazy and used a calculator)
    We take the Area we got from the trapezoid and add .5 to it because P(X>7) takes up half of the isosceles triangle.
    In the end we get 0.65625.

    Sunny Mei has earned ‘x’ amount of participation grade.



    I got something different.
    Height(H) = 0.5 = 1/2 *6H = 1/6
    From 3 to 15 there is 12, so H is in 9. 9/H = 7/h , h= 7/54 = 0.1296
    So, P(X>7) = A = 1/2*4*7/54 = 7/27 = 0.2592 I’m not sure.


    Ezra Halleck

    I think the confusion in the problem is perhaps the picture. I have attached the picture. Now that you have the picture, perhaps someone can come up with a solution. The best approach is to think complement. Find the area of the triangle to the left of 7 and then subtract from 1.

    Sunny’s approach is valid, but more complicated. I also believe that there is an error in the similar triangle calculation. The lengths for the known part of the triangles are 4 and 6 respectively.



    So then A= 1/2(2)(1/9) = 1/9
    p(x>7) = 1 – 1/9 = 0.88 , but how do we get 1/9? for h




    Well to Help you find how to get 1/9 Luis, you had to get the length of smaller part of the triangle which is 4 and then the bigger triangle which is 6, which becomes 4/6. Since we know the peak of the bigger triangle we get (H/(1/6)), Set them = to each other 4/6=(H/(1/6) which gives you 2/3=6H do the reciprocal and get 2/18, simplify and BAM!!!! you get 1/9.

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