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Advanced Math/Confidence Intervals of a Triangular distribution
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Question
QUESTION: Hi, I am trying to find a way to quickly calculate
confidence intervals using standard deviation in a
triangular probability distribution. For example I know
that 95% confidence level in a normal distribution is +-
2 standard deviations. I can't find out how to calculate
this at the 90%, 95% or 99% for a triangular distribution.
Any help will be greatly appreciated.
Thanks
Michael
ANSWER: I will take the distribution to be b(a-|x|) for x between -a and a.
To find out what b is, we need to find the area.
To do that, integrate a-x on 0 to a and multiply by 2 since its that same on both sides of 0. The integral is ax - x^2/2.
F(a)-F(0) is a^2/2 - 0 = a^2/2, so twice this value is a^2, so
b = 1/a^2.
To find the confidence interval, we want the integral from -c to c to be these values. That means we want (ax - x^2/2)/a^2 to be equal to 0.90, 0.95, and 0.98 when evaluated from -c to c.
This means that we want the integral of the function from 0 to c to be 0.9/2, 0.95/2. and 0.98/2. These values, respectively, are
0.45, 0.475, and 0.49.
---------- FOLLOW-UP ----------
QUESTION: Hi,
Thank you for the answer. It did lose me a bit though, am I
right in thinking that I would use the SD multiplied by
0.45,0.475 and 0.49? I don't think I am right in thinking
that.
Thanks
MIchael
Remember I had stated that we wanted to evaluate (ax - x^2/2)/a^2 to be equal to 0.90, 0.95, and 0.98 when evaluated from -c to c.
I simplified it a tad bit by saying we have to solve for x when c is 0.45, 0.475, and 0.95. That would be solving (ax - x²/2)/a² { yes, for awhie there, I couldn't use that special character ², but had to use ^2 } as I was saying, that would be solving for x in
(ax - x²/2)/a² = 0.45, (ax - x²/2)/a² = 0.475, and
(ax - x²/2)/a² = 0.49.
To do this, you need to know the value for a.
Maybe you would just get an answer in terms of a for all three.
The all will convert to a quadratic equation.
The x² coefficient is -1/(2a²). The x coefficient is 1/a. The value of the constant is -0.475.
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