1.

Find k ,such that the function `P(x)={{:(k({:(4),(x):}),,x=0","1","2","3","4,kgt0),(0,"otherwise."):}` is a probability mass function (p.m.f.)

Answer» Given , `P(x){{:(K((4)/(x))","x=0","1","2","3","4,kgt0,),(0" "", " "otherwise",):}`
`thereforeP[X=0]=K({:(4),(0):})=Kxx.^(4)C_(0)=K(1)=K`
`P[X=1]=K({:(4),(1):})=Kxx.^(4)C_(1)=K(4)=4K`
`P[X=2]=K({:(4),(2):})=Kxx.^(4)C_(2)`
`=K((4xx3)/(1xx2))=6K`
`P[X=3]=K({:(4),(3):})=Kxx.^(4)C_(3)=K(4)=4K`
`P[X=4]=K({:(4),(4):})=Kxx.^(4)C_(4)=K(1)=K`
Since `P[X=x]` is p . m .f., `sum_(x=0)^(4)P[X=x]=1`
`thereforeP[X=0]+P[X=1]+P[X=2]+P[X=3]+P[X=4]=1`
`thereforeK+4K+6K+4K+K=1`
`therefore16K=1`
`thereforeK=(1)/(16)`


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