Probability Generating Function

Question 1

Y is said to be the sum of two independent observations of X.
Which is true,

Y=X₁+X₂
Or
Y=2X

Answer 1

Both are correct
Since X observations are independent of each other, the random variable of X is the same at all times.
Hence, Y=2X

Question 2

Y=aX+b
G_y(t)=….?

Answer 2

G_y(t)=bG_x(t^a)

Question 3

Y=2X
Find G_y(t) using the correct method

Answer 3

Do NOT use X=Y/2

Y=2X→Y=X+X
G_y(t)
= G_x+x(t)
= G_x(t) • G_x(t)
= [G_x(t)]²