Let X i be a random variable distributed as N ( i , i 2 ) , i = 1 , 2 , 3. As-sume that the random variables X 1 , X 2 and X 3 are independent. Using only the three random variables X 1 , X 2 and X 3 give an example of a statistic that has a t distribution with two degrees of freedom.

Question

Let   X  i be a random variable distributed as   N  (  i  ,      i    2    )  ,  i  =  1  ,  2  ,  3. As-sume that the random variables       X    1    ,      X    2   and       X    3   are independent. Using only the three random variables       X    1    ,      X    2   and       X    3   give an example of a statistic that has a   t distribution with two degrees of freedom.

Raven Hawkins · Accepted Answer

First, numerator and denominator are not independent.Second, in denominator   (      X    1    +      X    2    ) is not chi-squared; worse it can take zero and negative values. So even if you achieved independence by using something like  (      X    2    +      X    3    )  ,, instead of   (      X    1    +      X    2    )  ,, the idea still wouldn&#039;t work.

Let X_i be a random variable distributed as N(i,i^2),i=1,2,3. As-sume that the random variables X_1,X_2, and X_3 are independent. Using only the three random variables X_1,X_2, and X_3 give an example of a statistic that has a t distribution with two degrees of freedom.

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