Write the values of `a` for which the following distribution of probabilities becomes a probability distribution: `X=x_i :\ -2 \ \ -1 \ \ 0 \ \ 1` `P(X=x_i):(1-a)/4 \ \ (1+2a)/4 \ \ (1-2a)/4 \ \ (1+a)/4 \ \ `

Answer

`-1/2 leq a leq 1/2`

Explanation

To have distribution of probabilities becomes a probability distribution, there are two conditions:


`1.`Sum of probability distribution is `1`, then,


`Sigma P(x_i)=P(X=-2)+P(X=-1
)+P(X=0)+P(X=1)`


`(1-a)/4 + (1+2a)/4 + (1-2a)/4 + (1+a)/4=1`


`1/4[1-a+1+2a+1-2a+1+a]=1/4[4]=1`


`2.` Another condition is that `0 leq P(x_i) leq 1`


Thus,


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