SOME SPECIFIC PROBABILITY DISTRIBUTIONS 2.2.4. Momen hi-square random variables Mean(x(u))=v= degrees of freedom (x2(u) (16) 2.3. The distribution function of x(v) s tabulated in most statistics and econometrics texts 2.4. Moment generating function. The moment generating function is as follows (1-2t) The first moment is E(x)=是(m=m)h t=0 (19) 3. THE STUDENTS T RANDOM VARIABLE This distribution was published by William Gosset in 1908. His employer, Guinness Breweries equired him to publish under a pseudonym, so he chose" Student 3. 1. Relationship of Students t-Distribution to Normal Distribution. The ratio t=xt0.1) has the Student's t density function with v degrees of freedom where the standard normal variate the numerator is distributed independently of the x variate in the denominator. Tabulations of the associated distribution function are included in most statistics and econometrics books note nat it is sy 3.2. Probability Density Function. The density of Student's t distribution is given b t2 √Dr(兰)SOME SPECIFIC PROBABILITY DISTRIBUTIONS 7 2.2.4. Moments of chi-square random variables. M ean (χ2 (ν)) = ν = degrees of freedom V ar (χ2 (ν)) = 2 ν Mode (χ2 (ν)) = ν − 2 (16) 2.3. The distribution function of χ2(ν). F(x; ν) = Z x 0 f (s; ν)ds (17) is tabulated in most statistics and econometrics texts. 2.4. Moment generating function. The moment generating function is as follows MX(t) = 1 (1 − 2 t) υ/2 ,t < 1 2 (18) The first moment is E ( X ) = d dt  1 ( 1 − 2 t ) υ/2  |t = 0 =  υ ( 1 − 2 t ) ( υ + 1)/2  |t = 0 = υ (19) 3. The Student’s t random variable This distribution was published by William Gosset in 1908. His employer, Guinness Breweries, required him to publish under a pseudonym, so he chose ”Student.” 3.1. Relationship of Student’s t-Distribution to Normal Distribution. The ratio t = N(0, 1) qχ2(ν) ν (20) has the Student’s t density function with ν degrees of freedom where the standard normal variate in the numerator is distributed independently of the χ2 variate in the denominator. Tabulations of the associated distribution function are included in most statistics and econometrics books. Note that it is symmetric about origin. 3.2. Probability Density Function. The density of Student’s t distribution is given by: f (t; ν ) = Γ ￾ ν + 1 2
√πν Γ ￾ ν 2
 1 + t 2 ν  −( ν +1) 2 − ∞ <t< ∞ (21)