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4.7: Chi-Squared Distributions

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  • In this section, we introduce the chi-squared distributions, which are very useful in statistics.

    Chi-Squared Distributions

    Definition \(\PageIndex{1}\)

    A random variable \(X\) has a chi-squared distribution with \(k\) degrees of freedom, where \(k\) is a positive integer, write \(X\sim\chi^2(k)\), if \(X\) has pdf given by
    $$f(x) = \left\{\begin{array}{l l}
    \displaystyle{\frac{1}{\Gamma(k/2)2^{k/2}} x^{k/2-1} e^{-x/2}}, & \text{for}\ x\geq 0, \\
    0 & \text{otherwise,}
    \end{array}\right. \notag$$

    Figure 1: Graph of pdf for \(\chi^2(1)\) distribution.

    The chi-squared distributions are a special case of the gamma distributions with \(\alpha = \frac{k}{2}, \lambda=\frac{1}{2}\), which can be used to establish the following properties of the chi-squared distribution.

    Properties of Chi-Squared Distributions

    If \(X\sim\chi^2(k)\), then \(X\) has the following properties.

    1. The mgf of \(X\) is given by
      $$M_X(t) = \frac{1}{(1-2t)^{k/2}},\quad \text{for}\ t<\frac{1}{2}\notag$$
    2. The mean of \(X\) is \(\text{E}[X] = k\), i.e., the degrees of freedom.
    3. The variance of \(X\) is \(\text{Var}(X) = 2k\), i.e., twice the degrees of freedom.

    Note that there is no closed form equation for the cdf of a chi-squared distribution in general. But most graphing calculators have a built-in function to compute chi-squared probabilities. On the TI-84 or 89, this function is named "\(\chi^2\)cdf''.

    The main applications of the chi-squared distributions relate to their importance in the field of statistics, which result from the following relationships between the chi-squared distributions and the normal distributions.

    Relationships of Chi-Squared Distributions

    1. If \(Z\) is a standard normal random variable, i.e., \(Z\sim N(0,1)\), then the distribution of \(Z^2\) is chi-squared with \(k=1\) degree of freedom.
    2. If \(X_1, \ldots, X_n\) is a collection of independent, chi-squared random variables each with 1 degree of freedom, i.e., \(X_i\sim\chi^2(1)\), for each \(i=1,\ldots,n\), then the sum \(X_1+\cdots+X_n\) is also chi-squared with \(k=n\) degrees of freedom.
    3. If \(X\sim\chi^2(k_1)\) and \(Y\sim\chi^2(k_2)\) are independent random variables, then \(X+Y\sim\chi^2(k_1+k_2)\).