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IAI Actuarial Core Principles · Actuarial Statistics · Random sampling and sampling distributions

Let X1,...,X5 be independent N(0,1) variables. Define T = X1 / sqrt((X2^2+X3^2+X4^2+X5^2)/4). Which statement about T is correct?

T has a t distribution with 4 degrees of freedom, because it is a standard normal divided by the square root of an independent chi-square with 4 degrees of freedom divided by 4.

  1. AT has a t distribution with 4 degrees of freedomCorrect
  2. BT has a t distribution with 5 degrees of freedom
  3. CT has an F distribution with 1 and 4 degrees of freedom
  4. DT has a standard normal distribution
  5. T has a chi-square distribution with 4 degrees of freedom

Explanation

The numerator is N(0,1), the denominator is the square root of an independent chi-square(4) divided by 4. This is the definition of t with 4 degrees of freedom. T^2 would be F(1,4), not T itself.

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