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where Z1, ..., Zk are independent normal variables, each having expected value 0 and variance 1. This distribution is usually written
If independent linear homogeneous constraints are imposed on these variables, the distribution of conditional on these constriants is , justifying the term "degrees of freedom". The characteristic function of the Chi-square distribution is
The chi-square distribution has numerous applications in inferential statistics, for instance in chi-square tests and in estimating variances. It enters the problem of estimating the mean of a normally distributed population and the problem of estimating the slope of a regression line via its role in Student's t-distribution. It enters all analysis of variance problems via its role in the F-distribution, which is the distribution of the ratio of two independent chi-squared random variables.
Its probability density function is
and pk(x) = 0 for x≤0. Here Γ denotes the gamma functionIn mathematics, the gamma function is a function that extends the concept of factorial to the complex numbers. Definition The notation Γ z is due to Adrien-Marie Legendre. If the real part of the complex number z is positive, then the integral : con. Tables of this distribution - usually in its cumulative form - are widely available (see the External Links below for online versions), and the function is included in many spreadsheetA spreadsheet is a rectangular table (or grid) of information, often financial information. It is, therefore, a kind of matrix. The word came from "spread" in its sense of a newspaper or magazine item (text and/or graphics) that covers two facing pages, es (for example MicrosoftMicrosoft Corporation , headquartered in Redmond, Washington, USA, is the world's largest software company (with over 50,000 employees in various countries, as of May 2004). Microsoft develops, manufactures, licenses and supports a wide range of software ExcelMicrosoft Excel is a spreadsheet program written and distributed by Microsoft for computers using the Microsoft Windows operating system and Apple Macintosh computers. It is overwhelmingly the dominant spreadsheet application available for these platforms) and all statistical packages.
If , then as tends to infinity, the distribution of tends to normality. However, the tendency is slow (the skewness is and the kurtosis is ) and two transformations are commonly considered, each of which approaches normality faster than itself:
Fisher showed that is approximately normally distributed with mean and unit variance.
Wilson and Hilferty showed in 19311931 is the common year starting on Thursday. see link for calendar) Events January January 4 Female aviator Elly Beinhorn begins her flight to Africa January 6 Thomas Edison submits his last patent application. January 22 Sir Isaac Isaacs sworn in as the that is approximately normally distributed with mean and variance .
The expected value of a random variable having chi-square distribution with k degrees of freedom is k and the variance is 2k. The median is given approximately by
Note that 2 degrees of freedom leads to an exponential distribution.
The chi-square distribution is a special case of the gamma distribution.