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• while some households may be more predisposed to

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Unformatted text preview: • While some households may be more predisposed to the particular service e.g. richer households may have a different p from poorer ones; this couldn’t be determined beforehand. With random sampling (so that one is likely to ask both rich and poor households) then the constant p represents an average over the whole population. (b) Use the binomial distribution tables to calculate: P( X = 4), P( X < 4), P( X ≥ 1). We have n=10, p=.2 ( = 4) = 0.0881 ( < 4) = ( ≤ 3) = 0.8791 ( ≥ 1) = 1 − ( = 0) = 1 − 0.1074 = 0.8926 3. A believer in the “random walk” theory of stock markets thinks that an index of stock prices has a probability of 0.65 of increasing in any one year. Let X be the number of years among the next 5 years in which the index rises. (a) What do we need to assume for X to have a binomial distribution. What are n and p ? What are the possible values that X can take? • Each of the n=5 years represent identical and independent trials. For this to be true the fact that the index went up last year does not change the probability it will go up in the subsequent year. • Index going up is denoted a success and all other outcomes (going down or staying unchanged)d are taken to be a failure. • The probability of a success p=0.65 is the same for each trial. (b) Assuming X has a binomial distribution, construct the probability distribution of X and draw the associated probability histogram. x 0 1 2 3 4 5 ( = ) 0.0053 0.0488 0.1811 0.3364 0.3124 0.1160 (c) What are the mean and variance of X ? ( ) = = 5 × 0.65 = 3.25 ( ) = (1 − ) = 5 × 0.65 × 0.35 = 1.1375 (d) Let Y be the number of years in which the index falls. Assuming X is binomial is Y binomial? What are the mean and standard deviation of Y ?...
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