1. In spelling bee competitions, contestants continue to compete until they spel
ID: 3361089 • Letter: 1
Question
1. In spelling bee competitions, contestants continue to compete until they spell a word incorrectly. Once they spell a word incorrectly, they are out of the competition. Suppose the probability that any word is misspelled is p. Let M be the number of the word on which a randomly selected speller goes out. For example, if a speller spells 4 words correctly and then misspells the 5th word, the observed value of M for that speller would be 5 (a) M follows a goometrie distribution. Explain why this is true, making sure to address all characteristics of the geometrie distribution, including identifying any appropriate parameter(s) (b) Let Xi,X2... . represent the number of letters in each word presented to a speller. Suppose the mean number of letters in a word is (i.e., E(X,) for all i) and the variance of the number of letters in a word is o? (i.e., Vir(%)-o? for all i). Consider TM-Xi + +Xy. i. In words, what does TMIAI m represent for a randomly selected speller? il. In words, what does Tar represent for a randomly selected speller? (e) Find E(TMIM m). (d) Find E(TM)Explanation / Answer
(a)
M follows the geometric distribution that gives the probability that the first occurrence of failure (spelling a word incorrectly) requires m independent trials, each with probability p. If the probability of failure on each trial is p, then the probability that the mth trial (out of m trials) is the first failure (misspells) is
Pr(M = m) = (1-p)m-1 p for m = 1,2,3
So, M ~ Geom(p) where p is the probability of misspelling a word in any trial
(b)
(i) TM | M = m represents the total number of letters of words in m trials given m is number of trials for a randomly selected speller until he/she misspells a word.
(ii) TM represents the total number of letters of words in number of trials for a randomly selected speller until he/she misspells a word.
(c)
E(TM | M = m) = E(X1 + X2 +..... + Xm) = E(X1) + E(X2) +..... + E(Xm)
= + + .... + = m
(d)
E(TM) = E(X1 + X2 +..... + XM) = E(X1) + E(X2) +..... + E(XM)
= + + .... + = M
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