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Math Help - geometric and Poisson distributions

  1. #1
    Newbie dangkhoa's Avatar
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    geometric and Poisson distributions

    I got the following two questions which I have no clue how to do it.

    (1)There are three coins in a barrel. These coins when flipped will come up heads with respective probabilities 0.3, 0.5 and 0.7. A coin is randomly selected from among these three coins and is then flipped until a head appears. Let N denote the number of flips necessary.

    What is the probability that N = n, n = 1, 2, 3, . . .?
    Is N a geometric random variable?

    Suppose now that each time a coin is flipped, it is returned to the barrel and a coin is then selected at random from the three coins, what is the distribution of N?

    (2) The number of fish that a fisherman catches in a day is a Poisson random variable with mean 30. On average the fisherman throws back two out of every three fish he catches.

    Find the probability that, on a given day, the fisherman takes home k fish, stating any independence assumptions that you make.

    Can anyone give me some hints please?

    Thank you so much
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by dangkhoa View Post
    I got the following two questions which I have no clue how to do it.

    (1)There are three coins in a barrel. These coins when flipped will come up heads with respective probabilities 0.3, 0.5 and 0.7. A coin is randomly selected from among these three coins and is then flipped until a head appears. Let N denote the number of flips necessary.

    What is the probability that N = n, n = 1, 2, 3, . . .?
    Is N a geometric random variable?
    Have you worked out P(N=n) for this situation? If so is there a p such that

    P(N=n)=(1-p)^n p?

    Which is what would be required for N to have a Geometric distribution.

    Suppose now that each time a coin is flipped, it is returned to the barrel and a coin is then selected at random from the three coins, what is the distribution of N?
    Now the probability of a head on each toss is the same so N has a geometric distribution. All you need do is calculate the probability of a head on a single toss.

    CB
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by dangkhoa View Post
    (2) The number of fish that a fisherman catches in a day is a Poisson random variable with mean 30. On average the fisherman throws back two out of every three fish he catches.

    Find the probability that, on a given day, the fisherman takes home k fish, stating any independence assumptions that you make.

    Can anyone give me some hints please?

    Thank you so much
    Assume that each fish is thrown back with probability 1/3 independently of what happens to any other fish.

    Then the number he takes home will have a Poisson distribution with mean 10.

    CB
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