pgf-hw - X n =3 ‡ z 2 · n-2 = z 2 ∞ X n =1 ‡ z 2 ·...

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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?
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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 G X ( z ) = z 2
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Unformatted text preview: X n =3 ‡ z 2 · n-2 = z 2 ∞ X n =1 ‡ z 2 · n = z 2 ± z 2 1-z 2 ¶ = z 3 2-z (3) To get the range and the probabilities, let’s just multiply it out: G X ( z ) = 1 z ± 1 3 z + 2 3 ¶ 4 = 1 z " ± 1 3 z ¶ 4 + 4 ± 1 3 z ¶ 3 ± 2 3 ¶ + 6 ± 1 3 z ¶ 2 ± 2 3 ¶ 2 + 4 ± 1 3 z ¶± 2 3 ¶ 3 + ± 2 3 ¶ 4 # = 1 81 z 3 + 8 81 z 2 + 24 81 z + 32 81 + 16 81 z-1 We can now read the information off 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...
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This note was uploaded on 07/26/2011 for the course MATH 530 taught by Professor Warren during the Summer '10 term at Ohio State.

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pgf-hw - X n =3 ‡ z 2 · n-2 = z 2 ∞ X n =1 ‡ z 2 ·...

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