Find value of a and b such that F(x) is a valid cumulative distribution function

In summary, the conversation discusses finding the values of a and b in order to make the given function a valid cumulative distribution function. The speaker mentions that the function should be right continuous and non-negative, but is unsure how to find the values of a and b. They ask for help and mention some potential restrictions on the values.
  • #1
chessmath
20
0
Hi
I have a question , the question asks find value of a and b such that F(x) is a valid cumulative distribution function?

1-a*exp(-x/b) x≥0
F(x)=
a*exp(x/b) x<0

My attempt to solve the problem:

I know F(x) when x goes to ∞ in 1 and when x goes to -∞ is 0. also I know F(x) should be right continuous and it is non-negative. However, non of them help me to find even one of the constants, any help will be appreciated ?
 
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  • #2


chessmath said:
Hi
I have a question , the question asks find value of a and b such that F(x) is a valid cumulative distribution function?

1-a*exp(-x/b) x≥0
F(x)=
a*exp(x/b) x<0

My attempt to solve the problem:

I know F(x) when x goes to ∞ in 1 and when x goes to -∞ is 0. also I know F(x) should be right continuous and it is non-negative. However, non of them help me to find even one of the constants, any help will be appreciated ?

If you want it to be continuous at x=0 what are the limits of both forms of F as x->0? Though actually if you only have right continuous, I don't think you have that restriction. Hmm. Not sure. I can see some restrictions on the values of a and b, but not how to calculate them.
 
Last edited:

FAQ: Find value of a and b such that F(x) is a valid cumulative distribution function

1. How do I find the values of a and b for a given cumulative distribution function?

The values of a and b can be found by solving the equations F(0) = 0, F(∞) = 1, and F(x) = ax + b for any x value. This will result in a system of linear equations that can be solved to find the values of a and b.

2. What is the significance of the values of a and b in a cumulative distribution function?

The values of a and b determine the shape and location of the cumulative distribution function. They impact the steepness of the curve and the starting and ending points on the x-axis, which can provide information about the probability distribution of a dataset.

3. Can the values of a and b be negative or zero in a cumulative distribution function?

Yes, the values of a and b can be negative or zero. However, if a is negative, the curve will be decreasing instead of increasing, and if b is negative, the curve will start below the x-axis. It is important to choose appropriate values for a and b to ensure a valid cumulative distribution function.

4. Is there a specific method for finding the values of a and b in a cumulative distribution function?

There is no specific method for finding the values of a and b in a cumulative distribution function. It often involves trial and error or using numerical methods like regression analysis to fit a curve to the data and determine the values of a and b.

5. How do I know if the values of a and b result in a valid cumulative distribution function?

To ensure that the values of a and b result in a valid cumulative distribution function, the function must satisfy the following conditions: F(x) is non-decreasing, F(0) = 0, F(∞) = 1, and F(x) is bounded by 0 and 1 for all x values. Additionally, the curve should have a smooth and continuous shape.

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