ECE109 Disc4.pdf - UC San Diego J Connelly ECE 109...

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UC San Diego J. Connelly ECE 109 Discussion 4 Notes Problem 4.1 Suppose we flip a coin. If the coin is heads, we randomly pick an integer from {- 2 , - 1 , 0 , 1 , 2 } . If the coin is tails, we randomly pick an integer from { 0 , 1 , 2 , 3 , 4 } . Let X be the integer that we pick. (a) What is the PMF of X ? (b) What is the expected value of X ? (c) What is the probability the coin was heads, assuming the number we picked was 2 ? (c) Suppose Y = X 2 . What is the PMF of Y ? Solutions (a) Let H be the event the coin was heads, then P ( X = k | H )= braceleftBigg 1 / 5 if k = - 2 , - 1 , 0 , 1 , 2 0 otherwise and P ( X = k | H c )= braceleftBigg 1 / 5 if k =0 , 1 , 2 , 3 , 4 0 otherwise so P ( X = k )= P ( X = k | H ) P ( H )+ P ( X = k | H c ) P ( H c )= 1 / 10 if k = - 2 , - 1 1 / 5 if k =0 , 1 , 2 1 / 10 if k =3 , 4 0 otherwise To verify that this is valid, we note that 4 summationdisplay k = - 2 P ( X = k )=1 . (b) We have E [ X ]= 4 summationdisplay k = - 2 k P ( X = k )= - 2 10 + - 1 10 + 1 5 + 2 5 + 3 10 + 4 10 =1 (c) P ( H | X =2)= P ( X =2 | H ) P ( H ) P ( X =2) = (1 / 5)(1 / 2) (1 / 5) =1 / 2 which implies H and { X = 2 } are independent events. However, if we know that X = 3 , then
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  • Spring '08
  • KennethZeger
  • Coin, 1 m, 1 k, UC San Diego, J. Connelly

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