Question

Suppose Y1, Y2, Y3, Y4, Y5 is a random sample from a gamma distribution where the shape parameter $\alpha$ is known to be 2 and the scale parameter $\beta$ is unknown.

a) Show that $\frac{Y_{3}}{\beta}$ is a pivotal quantity.

b) Show that $\sum_{i=1}^{5}\frac{Y_{i}}{\beta}$ is a pivotal quantity.

Answer 1

Solna We know a statistire random vareable is a pivotal quatity if ets distibution is free from para meters. Here, y ...Y are random samples from a gamma distribution where the shape parameter 222 and scale parameter ß is unknown. Then, Ex. (i) e o Jio, po BT2 ty (33)- 1 year 9,30 ; B70 Y3 I é Ě » 770 870 B? Now, let Ta - Bl and a B also to Sence, g de ses monotone in 70, then wring transformation melhod. f (d) - Fy (Bt). 1 7 sto 2 - Bte f ; to d - tēt; to. which is again gamma distributson with scale para meter en known equal tot shape parameter au 2 is. Growen also renee, T2 T3 has distribution in which parameters are known as Preenet, & in a pirstal quantity

& here we have proved that 23 o Camma (2, 1) TTTTT Similarly, we can prove that Yi, le, Yy 18 will also B B B 'ß follow gamma distributson with 2=2, Bafil. gamma (2, 13 and we also know that sum of ne lid gamma random variables with parameter & and ß is also a gamma distributsen weilt parameters na and ß. vence Eli will follow gamma (10, 1) which is also known dists buts on. Kence, & fe is also a pirstal quantity