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# S1 Help - Cumulative distribution function

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1. 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?
2. (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?
represents the cumulative distribution function, its range is given as

So to solve we know that at the lowest point of range, , and at the highest point,

The represents the total probability(the total area under the p.d.f graph)
3. (Original post by raheem94)
represents the cumulative distribution function, its range is given as

So to solve we know that at the lowest point of range, , and at the highest point,

The 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?

4. (Original post by Ryan44)
So does F(y) always have a range of 0 < y < 1?

No, it depends upon the question.

Example:
It has a range of . Then
5. (Original post by raheem94)
No, it depends upon the question.

Example:
It has a range of . Then
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?
6. (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
7. (Original post by raheem94)
Yes.

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

Do you know the difference between
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?
8. (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 curve is 1, so if its range is
Then the total area is

Hope it makes sense.

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