Question

What is the chi-square distribution? Explain your answer?

Answer 1

The chi-square distribution was first discovered by Helmert in 1876 and later independently by Karl Pearson in 1900.

If X is N(0,1) variate, then X² is known as the Chi-square variate.

If X~ N() then the standard normal deviate Z = ~ N(0,1) and Z² is distributed as Chi-square () with 1 degree of freedom (d.f.).

If X₁, X₂, ..., X_n are n independent variates distributed as N(), then

is distributed as Chi-square with n d.f.

The probability density function of the Chi-square distribution is,

,

.

Properties of the Chi-square distribution:

(i) Mean is n.

(ii) Variance is 2n.

(iii) Mode lies at the point .

(iv) Moment generating function is (1-2t)^-n/2.

(v) Characteristic function is (1-2it)^-n/2.

(vi) Measure of skewness, i.e. positively skewed.

(vii) Measure of kurtosis, i.e. leptokurtic.

(viii) Chi-square distribution is used to test whether a hypothetical value of the population variance is true or not.

(ix) is used to make a test of goodness of fit.

(x) It is used to test the independence of attributes.

(xi) It is used to test the validity of a hypothetical ratio.

(xii) It is used to test the homogeneity of several population variances.

(xiii) It is used to test the equality of several population correlation coefficients.