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Thread: Random Variable of two other random variables. Confusion to the max!

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    Random Variable of two other random variables. Confusion to the max!




    Here is the question at hand:
    I don't even know how to begin this question. I'm not looking for a complete solution per se but I don't even know where to begin.
    Suppose that X is a binomial random variable with parameters (m, θ) and Y is another
    binomial random variable with parameters (n, θ) and suppose that X and Y are independent.
    Let Z = X+Y . Show that Z is another Binomial random variable with parameters (m+n, θ).
    Hint:
    P
    Note that you have to find the PMF of Z: P(Z = z). You may use the fact that
    sum over k of (n choose k)(n choose (z-k) = (m plus n) choose z where the sum ranges over k for which the binomial coefficients are
    defined.

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    Re: Random Variable of two other random variables. Confusion to the max!


    The technique in a general mathematical context is called convolution.

    In probability context, you can understand it as law of total probability.

    The first key idea is when Z = z, you need to know/list all possible pairs of (X, Y) such that X + Y = z.

    For example, when Z = 2, the possible pairs are (0, 2), (1, 1), (2, 0), provided that m, n > 1.

    Try to generalize a bit and see if you can apply the law of total probability to solve that.

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