How do you transform Gamma to Chi-squared distribution

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This question likely needs a homework or self study tag and the OP, who has asked several similar kinds of questions before, should know better than to skip this.

Here are some hints:

  • The sum of independent Gamma random variables with identical scale parameter is a Gamma random variable with the same scale parameter and order parameter equal to the sum of the order parameters. So you should be able to show that $\sum_i X_i$ is a Gamma random variable.

  • If $E[Y] = \mu$, then $E[\mu^{-1}Y] = 1$. Furthermore, if $Y$ is a Gamma random variable, then so is $aY$ a Gamma random variable. You should be able to figure out the distribution of $2\theta\sum_i X_i$ from this.

  • A Gamma random variable with order parameter $\frac{n}{2}$ and mean $n$ is a $\chi^2$ random variable with $n$ degrees of freedom.

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Updated on December 07, 2020

Comments

  • moku
    moku almost 3 years

    Here is the question not sure how to turn a Gamma into a Chi-Squared:

    Suppose $X_1....X_n$ is a sample from the distribution Gamma($\alpha=3,\ \lambda=\theta$) with unknown $\theta > 0$. We wish to eventually construct a 98% Confidence Interval for the scale parameter $\theta$

    a) Show first that $2\theta\sum_{1 \le \ i \le n} X_i$ has Chi-squared distribution, identify the number of degree of freedom

    b) Using $2\theta\sum_{1 \le \ i \le n} X_i$ as a piviot, explain what tables you would need to use in order to find an $a(x_1,...x_n)$ and a $b(x_1,...x_n)$ so that: $$P(a(x_1,...x_n) < \theta < b(x_1,...x_n))=.98$$ The 98% CI for $\theta$ is $a(x_1,...x_n),b(x_1,...x_n)$

    • Frank
      Frank over 9 years
      If you are asking how to prove part (a), then what you should do is look up the moment generating functions or characteristic functions for Gamma and Chi-Squared distributions; look up the result that says that the MGF or CF of a sum of independent RV's is the product of the MGF's and CF's of those RV's; and then recall the uniqueness theorem for MGF's and CF's.
    • moku
      moku over 9 years
      Ok I'll take go look at the wiki of gamma distributions