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S1 Help - Cumulative distribution function

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    Hi there,

    Just a quick question, I am slightly confused as to the mark scheme for this question:

    (b) The continuous random variable Y takes values between 1 and 2 and its cumulative distribution function F is given, for 1 < y < 2, by
    F(y ) = ay + by^2.
    Find the values of the constants a and b.

    Now I understand to input F(1) into it so I get a+b=0 (like what the mark scheme says) and then F(2) so 2a+4b.. but then it equals this second equation to 1 for some reason?

    I know how to find a and b after getting the two equations but I just wanted to know why that second equation equals 1?
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    (Original post by Ryan44)
    Hi there,

    Just a quick question, I am slightly confused as to the mark scheme for this question:

    (b) The continuous random variable Y takes values between 1 and 2 and its cumulative distribution function F is given, for 1 < y < 2, by
    F(y ) = ay + by^2.
    Find the values of the constants a and b.

    Now I understand to input F(1) into it so I get a+b=0 (like what the mark scheme says) and then F(2) so 2a+4b.. but then it equals this second equation to 1 for some reason?

    I know how to find a and b after getting the two equations but I just wanted to know why that second equation equals 1?
     F(y) represents the cumulative distribution function, its range is given as  0 &lt; y &lt;  1

    So to solve we know that at the lowest point of range,  F(1) = 0 , and at the highest point,  F(2) = 1

    The  1 represents the total probability(the total area under the p.d.f graph)
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    (Original post by raheem94)
     F(y) represents the cumulative distribution function, its range is given as  0 &lt; y &lt;  1

    So to solve we know that at the lowest point of range,  F(1) = 0 , and at the highest point,  F(2) = 1

    The  1 represents the total probability(the total area under the p.d.f graph)
    So does F(y) always have a range of 0 < y < 1?

    Thanks for your answer!
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    (Original post by Ryan44)
    So does F(y) always have a range of 0 < y < 1?

    Thanks for your answer!
    No, it depends upon the question.

    Example:
    It  F(y) has a range of  2 &lt; y &lt; 5 . Then  F(2)=0 \ and \ F(5)=1
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    (Original post by raheem94)
    No, it depends upon the question.

    Example:
    It  F(y) has a range of  2 &lt; y &lt; 5 . Then  F(2)=0 \ and \ F(5)=1
    I see what you mean as in my question it is between 1 < y < 2 and that F(1)=0 and F(2)=1

    But does the F(higher boundary) always equal to 1 and the F(lower boundary) always = to 0?
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    (Original post by Ryan44)
    I see what you mean as in my question it is between 1 < y < 2 and that F(1)=0 and F(2)=1

    But does the F(higher boundary) always equal to 1 and the F(lower boundary) always = to 0?
    Yes.

    Do you understand what does F(y) actually represent?

    Do you know the difference between  f(y) \ and \ F(y) \ ?
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    (Original post by raheem94)
    Yes.

    Do you understand what does F(y) actually represent?

    Do you know the difference between  f(y) \ and \ F(y) \ ?
    Couldn't give you the definition between the two although I know how to convert f(y) to F(y) and vice versa correctly.

    f(y) to F(y) you integrate? and F(y) to f(y) you differentiate the function?
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    (Original post by Ryan44)
    Couldn't give you the definition between the two although I know how to convert f(y) to F(y) and vice versa correctly.

    f(y) to F(y) you integrate? and F(y) to f(y) you differentiate the function?
    Yes, you are correct. I thought that you might be misunderstanding them.

    We know the total area under the  f(y) curve is 1, so if its range is  0 &lt; y &lt; 3
    Then the total area is  \displaystyle \int ^3 _0 f(y) \ dy = 1 = F(3)

    Hope it makes sense.

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