please solve and show all steps 3. Audrey, an astronomer is searching for extra-

Question

please solve and show all steps

3. Audrey, an astronomer is searching for extra-solar planets using the technique of relatriste lensing number ofplanets that can be found this way, actually finding one takes time and luck; and finding one planet does not help at all with finding planets of other stars in the same part of the sky. Audrey is good at it, and finds one planet at a time, on average once every three months. Though there are believed to be a very large a) Find the expected value and standard deviation of the number of planets she will find in the next two years. b.) When she finds her sixth new planet, she will be eligible for a prize. Find the expected valueof the amount of time until she is eligible for that prize. c.) Find the probability that she will become eligible for that prize within one year.

Explanation / Answer

Here the given problem is of Poisson distribtuion.

Here audrey find one planet in 3 months on average

(a) Expected number f plantes found by Audrey in next 2 years = 24/3 = 8 planets

standard deviation of the number of planets found be Audrey in next 2 years = sqrt(8) = 2.8284 years

(b) Here she finds one planet in 3 months on average.

so she will find her sixth planet in 6 * 3 = 18 months or say 1.5 years.

(c) Now we have to find that she will be eligible for that prize withing one year. that means she will get her prize in 12 months when 18 months are the expected time to get it.

Pr(t ? 12 months) = Poisson (t ? 12 months, 18 months) = 0.092

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