a. Differentiate the series 1 + x + x² + ·.. + x" +.. х to obtain a series for 1/(1 – x)². b. In one throw of two dice, the probability of getting a roll of 7 is p = 1/6. If you throw the dice repeatedly, the probability that a 7 will appear for the first time at the nth throw is q"-'p, where q = 1 – p = 5/6. The expected number of throws un- til a 7 first appears is Enq"-'p. Find the sum of this series. c. As an engineer applying statistical control to an industrial operation, you inspect items taken at random from the as- sembly line. You classify each sampled item as either “good" or “bad." If the probability of an item’'s being good is p and of an item's being bad is q = 1 – p, the probability that the first bad item found is the nth one inspected is p"-'q. The average number inspected up to and including the first bad item found is E-¡np"-'q. Evaluate this sum, assuming 0

Calculus: Early Transcendentals
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Author:James Stewart
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Chapter1: Functions And Models
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Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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a. Differentiate the series
1 + x + x² + ·.. + x" +..
х
to obtain a series for 1/(1 – x)².
b. In one throw of two dice, the probability of getting a roll of 7
is p = 1/6. If you throw the dice repeatedly, the probability
that a 7 will appear for the first time at the nth throw is q"-'p,
where q = 1 – p = 5/6. The expected number of throws un-
til a 7 first appears is Enq"-'p. Find the sum of this series.
c. As an engineer applying statistical control to an industrial
operation, you inspect items taken at random from the as-
sembly line. You classify each sampled item as either “good"
or “bad." If the probability of an item’'s being good is p and
of an item's being bad is q = 1 – p, the probability that
the first bad item found is the nth one inspected is p"-'q.
The average number inspected up to and including the first
bad item found is E-¡np"-'q. Evaluate this sum, assuming
0 <p < 1.
Transcribed Image Text:a. Differentiate the series 1 + x + x² + ·.. + x" +.. х to obtain a series for 1/(1 – x)². b. In one throw of two dice, the probability of getting a roll of 7 is p = 1/6. If you throw the dice repeatedly, the probability that a 7 will appear for the first time at the nth throw is q"-'p, where q = 1 – p = 5/6. The expected number of throws un- til a 7 first appears is Enq"-'p. Find the sum of this series. c. As an engineer applying statistical control to an industrial operation, you inspect items taken at random from the as- sembly line. You classify each sampled item as either “good" or “bad." If the probability of an item’'s being good is p and of an item's being bad is q = 1 – p, the probability that the first bad item found is the nth one inspected is p"-'q. The average number inspected up to and including the first bad item found is E-¡np"-'q. Evaluate this sum, assuming 0 <p < 1.
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