Statisticians and mathematicians often measure failure rates with a probability density function called the Weibull distribution. The equation of the Weibull distribution is p(x) = a(b^-a)(x^(a-1))e^-(x/b)^a, where b is a scaling parameter, a is the shape parameter and e is the number 2.1828.
In the Weibull distribution, the variance, V, is a function of the parameters a and b. The variance formula is V = b^2*(G(1 + 2/a) - G(1 + 1/a)^2), where G denotes the Gamma function. To compute values of the Gamma function, you must use a graphing calculator or mathematical software.
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Things you need
- Graphing calculator or mathematical software
Identify the parameter a in the given Weibull distribution. For example, if you have the equation p(x) = 4(3^-4)(x^(4-1))e^-(x/3)^4, then a = 4. The value of a is the last exponent in the equation.
Identify the parameter b in the given Weibull distribution. In the equation p(x) = 4(3^-4)(x^(4-1))e^-(x/3)^4, the value of b is 3. The parameter b is the denominator of x in the fraction x/3 that occurs at the end of the expression.
Determine G(1 + 2/a) using your value of a and software that can compute Gamma functions. Using a = 4, you must calculate G(1.5). Using a calculator or software, you obtain G(1.5) = 0.886227.
Compute G(1 + 1/a)^2 using the same value of a. For instance, since a = 4, you must find G(1.25)^2, which equals 0.821565
Subtract the value you obtained in Step 4 from the value you obtained in Step 3. For example, you calculate 0.886227 - 0.821565 = 0.064662.
Multiply this number by b^2. This is the variance of the Weibull distribution. For instance, since b = 3, you compute (3^2)*(0.064662) = 0.581958 as the variance.
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