{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# pgf-hw - X n =3 â€ z 2 Â n-2 = z 2 âˆž X n =1 â€ z 2 Â n =...

This preview shows pages 1–2. Sign up to view the full content.

A few basic questions on probability generating functions 1 Let X be a random variable with the following mass function: P [ X = - 2] = 1 3 P [ X = 3] = 1 6 P [ X = π ] = 1 8 P X = 7 2 = 3 8 . Find the probability generating function G X ( z ). 2 Suppose P [ X = n ] = 1 2 n - 2 for n = 3 , 4 , 5 , . . . . Find a compact formula for the probability generating function G X ( z ). 3 Suppose G X ( z ) = ( 1 3 z + 2 3 ) 4 z . What is the range of X ? What is the mass function?

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
A few basic questions on probability generating functions Page 2 of 2 Solutions (1) Remember that your output values become powers on z , and your probabilities become coefficients of those powers of z . G X ( z ) = 1 3 z - 1 + 1 6 z 3 + 1 8 z π + 3 8 z 7 / 2 . (2) By definition, G X ( z ) = X n Range ( X ) P [ X = n ] z n . Here, we have the range and also the probabilities, so G X ( z ) = X n =3 1 2 n - 2 z n . Factoring out a z 2 to even up the powers inside, we get
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: X n =3 â€¡ z 2 Â· n-2 = z 2 âˆž X n =1 â€¡ z 2 Â· n = z 2 Â± z 2 1-z 2 Â¶ = z 3 2-z (3) To get the range and the probabilities, letâ€™s just multiply it out: G X ( z ) = 1 z Â± 1 3 z + 2 3 Â¶ 4 = 1 z " Â± 1 3 z Â¶ 4 + 4 Â± 1 3 z Â¶ 3 Â± 2 3 Â¶ + 6 Â± 1 3 z Â¶ 2 Â± 2 3 Â¶ 2 + 4 Â± 1 3 z Â¶Â± 2 3 Â¶ 3 + Â± 2 3 Â¶ 4 # = 1 81 z 3 + 8 81 z 2 + 24 81 z + 32 81 + 16 81 z-1 We can now read the information oï¬€ the function. The range is {-1 , , 1 , 2 , 3 } and the mass function is: P [ X =-1] = 16 81 P [ X = 0] = 32 81 P [ X = 1] = 24 81 P [ X = 2] = 8 81 P [ X = 3] = 1 81 Last compiled at 9:28 P.M. on July 12, 2011...
View Full Document

{[ snackBarMessage ]}