Question

Random samples Y_1,Y_2,Y_3,\dots,Y_nY1,Y2,Y3,…,Yn were selected from an X distribution with mean µ and variance σ2.

a) If Z_1=(Y_1-μ)/σZ1=(Y1−μ)/σ, state the distribution for Z1 . (10 marks)

Let

2/2 = ***)*,272 = 4**3)*,732 = 1**) ....Z,= (**)*,

i. Can we conclude that the distribution answer in 1a) is same with the distribution for Z_1^2Z12 Justify your answer. (10 marks)

ii. If Z_1^2+Z_2^2+Z_3^2+⋯+Z_n^2Z12+Z22+Z32+⋯+Zn2 = W, can we assume W has a similar distribution as X. Explain your answer. (10 marks)

Answer 1

Yo were Random samples Y , Yao.. selected from x distribution with mean and variance ? an M 2,= "Y,- M Ya-M Yo-M Z 2. Zn= MaM Mod +62) = 467.5 M) = Edu)-cm var(z) = vas (1,6mm) - var(40) - varles 57 02 pe E Z, NNCOD Z; follow the standard normal distribution with mean M2=0 and standard deviation ej 2, MN (0,0) 2,2 = (mat å 2 Zi follow Chir squase distribution with a degree of freedom. In The rerefore, the distribution answer in la) is not same with alhe distribution for Z n = Wa 2 2 Chuyo wan 2 +2...+ 22 awa Hence, w does not similas distribution with distribution of x. have