Let X have a Weibull distribution. Verify that μ=βΓ(1+1 / α) .[Hint: In the integral for E(X) )

step 1 From the textbook, we are given the formula for the mean of a Wibo distribution. In this exercis

Let X have a Weibull distribution. Verify that μ=βΓ(1+1 / α)...

Let $X$ have a Weibull distribution. Verify that $\mu=\beta \Gamma(1+1 / \alpha) .[$Hint$:$ In the integral for $E(X)$ ) make the change of variable $y=(x / \beta)^{\alpha},$ so that $x=\beta y^{1 / \alpha} . ]$

Key Concepts

Weibull Distribution

The Weibull distribution is a continuous probability distribution commonly used in reliability engineering and failure analysis, among other fields. It is characterized by its scale and shape parameters, and its probability density function is formulated in a way that allows a variety of data shapes, making it versatile for modeling time-to-failure phenomena.

Expected Value

The expected value, or mean, of a random variable is a fundamental concept in probability that represents the long-run average outcome of a stochastic process. It is calculated by integrating the product of the variable's value and its probability density function over the range of possible values.

Gamma Function

The Gamma function extends the concept of factorials to non-integer values and plays a central role in many areas of mathematical statistics. In the context of the Weibull distribution, it is used to evaluate integrals that arise when calculating moments, specifically converting the integral into one that resembles the Gamma function.

Change of Variables in Integration

The change of variables, or substitution method, is an essential technique in calculus used to simplify the evaluation of integrals. It involves substituting a new variable for an existing one to transform the integrand into a more workable form, often matching it to known integral formulas like that of the Gamma function.

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