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The simplest chi-squared distribution is the square of a standard normal distribution. So wherever a normal distribution could be used for a hypothesis test, a chi-squared distribution could be used. Suppose that Z {\displaystyle Z} is a random variable sampled from the standard normal distribution, where the mean is 0 {\displaystyle 0} and the ...
The square of a standard normal random variable has a chi-squared distribution with one degree of freedom. If X is a Student’s t random variable with ν degree of freedom, then X 2 is an F (1,ν) random variable. If X is a double exponential random variable with mean 0 and scale λ, then |X| is an exponential random variable with mean λ.
It is the distribution of the positive square root of a sum of squared independent Gaussian random variables. Equivalently, it is the distribution of the Euclidean distance between a multivariate Gaussian random variable and the origin. The chi distribution describes the positive square roots of a variable obeying a chi-squared distribution.
Here is one based on the distribution with 1 degree of freedom. Suppose that X {\displaystyle X} and Y {\displaystyle Y} are two independent variables satisfying X ∼ χ 1 2 {\displaystyle X\sim \chi _{1}^{2}} and Y ∼ χ 1 2 {\displaystyle Y\sim \chi _{1}^{2}} , so that the probability density functions of X {\displaystyle X} and Y ...
We've assumed, without loss of generality, that , …, are standard normal, and so + + has a central chi-squared distribution with (k − 1) degrees of freedom, independent of . Using the poisson-weighted mixture representation for X 1 2 {\displaystyle X_{1}^{2}} , and the fact that the sum of chi-squared random variables is also a chi-square ...
Chi-squared distribution, showing χ 2 on the x-axis and p-value (right tail probability) on the y-axis.. A chi-squared test (also chi-square or χ 2 test) is a statistical hypothesis test used in the analysis of contingency tables when the sample sizes are large.
Some important special cases of this have all weights of the same sign, or have central chi-squared components, or omit the normal term. Since a non-central chi-squared variable is a sum of squares of normal variables with different means, the generalized chi-square variable is also defined as a sum of squares of independent normal variables ...
By the normal approximation to a binomial this is the squared of one standard normal variate, and hence is distributed as chi-squared with 1 degree of freedom. Note that the denominator is one standard deviation of the Gaussian approximation, so can be written