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Question

Find the mean variance and standard deviation of the following probability distribution
xi : a b
pi : p q
where p + q = 1

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Solution

xi pi pixi pixi2
a p ap a2p
b q bq b2q
pixi = ap + bq pixi2= a2p + b2q

Now,Mean=pixi=ap+bqVariance=pixi2-Mean2=a2p+b2q-ap+bq2 =a2p+b2q-a2p2-b2q2-2abpq =a2p-a2p2+b2q-b2q2-2abpq =a2p1-p+b2q1-q-2abpq =a2pq+b2qp-2abpq p+q=1 =pqa2+b2-2ab =pqa-b2Step Deviation=Variance =pqa-b2 =a-bpq

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