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Statistics help please: Linear Combinations of Random Variable Watch

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    The independent variables X and Y are such that X~B(10,0.8) and Y~Po(3). Find P(2X - Y =18).
    I have tried to calculate the mean and variance of the combined variables but then realized that there is not much I can do with them since the final distribution is not a normal distribution. Please advise.
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    (Original post by VTan)
    The independent variables X and Y are such that X~B(10,0.8) and Y~Po(3). Find P(2X - Y =18).
    I have tried to calculate the mean and variance of the combined variables but then realized that there is not much I can do with them since the final distribution is not a normal distribution. Please advise.
    Have you covered the convolution formula for the sum of two discrete random variables in your course? This gives you, if  Z = X + Y,

    \displaystyle \mathbb{P}(Z = z) = \sum_{x \in Im(X)} \mathbb{P}(X=x, y=z-x)

    Now adapt the proof of this to the sum 2X - Y and do the resulting calculation.
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    (Original post by Gregorius)
    Have you covered the convolution formula for the sum of two discrete random variables in your course? This gives you, if  Z = X + Y,

    \displaystyle \mathbb{P}(Z = z) = \sum_{x \in Im(X)} \mathbb{P}(X=x, y=z-x)

    Now adapt the proof of this to the sum 2X - Y and do the resulting calculation.
    I don't think that's under the CIE syllabus exam board. I haven't seen that formula before. It's alright, I'll try to read up on it and see how it works. Thank you for the help! =)
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    (Original post by VTan)
    I don't think that's under the CIE syllabus exam board. I haven't seen that formula before. It's alright, I'll try to read up on it and see how it works. Thank you for the help! =)

    Ah! Is this at "A" level? Then we can probably get away with something less heavy. You're told that X is binomial with a range of values from 0 to 10. Y is Poisson with a range of values from zero upwards. You want the value of P(Z = 2X - Y = 18). What values of X and Y will give you Z = 18? There are two possibilities and they are independent.
 
 
 
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