Skip to content

FRM Part I · FRM Exam Part I · Common Univariate Random Variables

Z1, Z2, Z3 and Z4 are independent standard normal variables. Let X = Z1² + Z2² + Z3² + Z4², and let Y = Z1 / √[(Z2² + Z3² + Z4²)/3]. Which statement about X and Y is correct, and what is Var(X)?

X is chi-squared with 4 degrees of freedom and its variance is 8, since a chi-squared variable has variance twice its degrees of freedom. Y is a standard normal over the root of an independent chi-squared with 3 df divided by 3, so it is Student's t with 3 degrees of freedom.

  1. AX is chi-squared with 4 degrees of freedom and Var(X) = 8; Y is Student's t with 3 degrees of freedomCorrect
  2. BX is chi-squared with 4 degrees of freedom and Var(X) = 4; Y is Student's t with 3 degrees of freedom
  3. CX is chi-squared with 4 degrees of freedom and Var(X) = 8; Y is Student's t with 4 degrees of freedom
  4. DX is chi-squared with 3 degrees of freedom and Var(X) = 6; Y is Student's t with 3 degrees of freedom

Explanation

A sum of 4 squared independent standard normals is chi-squared with k = 4; its variance is 2k = 8 (the mean is k = 4). Y is a standard normal divided by the root of an independent chi-squared with 3 df divided by 3, so it is t with 3 df. Variance 4 confuses the variance with the mean.

Did you get it right without looking?

One question tells you little. A timed set on Common Univariate Random Variables shows your real accuracy, how long you take and where you lose marks.

Practise Common Univariate Random VariablesFree diagnostic mock

More Common Univariate Random Variables questions