1 Moments of the Bivariate T-Distribution Anwar H. Joarder Department of Mathematical Sciences King Fahd University of Petroleum and Minerals Dhahran 31261, Saudi Arabia Email: anwarj@ kfupm.edu.sa Abstract The probability density function of the bivariate t-distribution can be represented by a scale mixture representation of the bivariate normal distribution with an 'inverted' chisquare distribution. The product moments of the bivariate t-distribution are derived by exploiting the scale mixture. Some standardized moments of the bivariate t-distribution are also derived. AMS Mathematics Subject Classifications: 60E10, 60-02 Key Words: Bivariate normal distribution, bivariate t-distribution, product moments, standardized moments 1. Introduction The bivariate t-distribution arises as a derived sampling distribution from the bivariate normal distribution and the chisquare distribution (Anderson, 2003, 289). In this paper we emphasize a scale mixture representation to calculate some product moments and standardized moments of the bivariate t-distribution. Because of the increase in the use of the multivariate t-distribution in business, especially, in stock returns, the paper will stimulate research in business, econometrics and statistics. Interested readers may go through Lange, Little and Taylor (1989), Billah and Saleh (2000), Kibria and Saleh (2000) and Kotz and Nadarajah (2004). (i) The Univariate T-Distribution Theorem 1.1 Let 2 ~ (0, ) Z N τ and 2 2 ~ dW ν ν χ − Τ where the symbol d means that both sides of it have the same distribution. Then 1/2 ( ) ( / ) ~ a W Z t ν ν , 2 /2 ( 1) /(2 ) 2( / 2) () () ( / 2) b h e ν ν ν τ ν τ τ ν − + − Τ = Γ , 0 τ < , (1.1) (c) the pdf of the univariate t-dsitribution with ν degrees of freedom has the following representation: 2 1 2 0 1 ( ) exp () . 2 2 z f z h d τ τ τ τ π ∞ Τ ⎛ ⎞ = − ⎜ ⎟ ⎝ ⎠ ∫ (1.2) Proof. The proof of the first two parts are well known while the third part follows by transformation 2 / w ν τ = with Jacobian 3/2 ( ) / 2. J w w τ ν − → =