91Ó°ÊÓ

X be the random variable that can take only two values a and b with\({\rm P}\left( {X = a} \right) = p\)and \({\rm P}\left( {X = b} \right) = 1 - p\)

We wish to express\(f\left( x \right)\)in the form\({p^{\alpha \left( x \right)}}{\left( {1 - p} \right)^{\beta \left( x \right)}}\)

Where\(\alpha \left( x \right) = 1\)and\(\beta \left( x \right) = 0\),and\(x = a\)and\(\alpha \left( x \right) = {\alpha _1} + {\alpha _1}x\)and

\(\beta \left( x \right) = {\beta _1} + {\beta _2}x\)For x=b

If we chose\(\alpha \left( x \right)\)and\(\beta \left( x \right)\)to be linear function of the form\(\alpha \left( x \right) = {\alpha _1} + {\alpha _1}x\)and

\(\beta \left( x \right) = {\beta _1} + {\beta _2}x\), then following pairs of equation must be satisfied.

\(\begin{aligned}{l}{\alpha _1} + {\alpha _2}a = 1\\{\alpha _1} + {\alpha _2}b = 0,\end{aligned}\)

And

\(\begin{aligned}{l}{\beta _1} + {\beta _2}a = 0\\{\beta _1} + {\beta _2}b = 1\end{aligned}\)

Hence,

\(\begin{aligned}{l}{\alpha _1} = - \frac{b}{{a - b}}\\{\alpha _2} = - \frac{1}{{a - b}}\end{aligned}\)

\(\begin{aligned}{l}{\beta _2} = - \frac{b}{{b - a}}\\{\beta _2} = - \frac{1}{{b - a}}\end{aligned}\)

Hence pf of x is \({p^{\alpha \left( x \right)}}{\left( {1 - p} \right)^{\beta \left( x \right)}}\)with the given values of \(\alpha \) and \(\beta \)