For each of the following, give the name of the distribution, the values of any
ID: 3131895 • Letter: F
Question
For each of the following, give the name of the distribution, the values of any parameters, a formula for the exact probability, and only if appropriate (a binomial distribution with large n and p close to 0 or 1), use the Poisson distribution to estimate the probability. There are 600 frogs in a swamp. On average, each frog leaps out of the swamp to catch a fly 16 times per 8-hour night. If I watch the swamp for one minute during the night, what's the probability that I see at least 16 frogs leaping for flies?Explanation / Answer
18.
Hence, a frog leaps every 8/16 = 0.5 hours or 30 minutes on the average.
Hence, on the average, we see 600 frogs every 30 minutes, or, 600/30 = 20 frogs per minute.
Hence, this is
i) a Poisson distirbution
ii) with mean u = 20 frogs per minute
iii)
Note that the probability of x successes is
P(x) = u^x e^(-u) / x!
where
u = the mean number of successes = 20
x = the number of successes
Hence,
P(x) = 20^x e^(-20) / x!
Hence,
P(at least 16) = 1 - Sum[P(x)]|(x=0 to 15) [ANSWER]
where P(x) is as above.
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Note that P(at least x) = 1 - P(at most x - 1).
Using a cumulative poisson distribution table or technology, matching
u = the mean number of successes = 20
x = our critical value of successes = 16
Then the cumulative probability of P(at most x - 1) from a table/technology is
P(at most 15 ) = 0.156513135
Thus, the probability of at least 16 successes is
P(at least 16 ) = 0.843486865 [ANSWER, PROBABILITY OF AT LEAST 16 FROGS]
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