In general, the expected value of an event X is given by the formula below
[tex]E\lbrack X\rbrack=\sum_ix_i*f(x_i)[/tex]
Therefore, in our case,
[tex]E\lbrack E\rbrack=1*P(1)+8*P(8)+32P(32)+64P(64)[/tex]
Thus,
[tex]\begin{gathered} \Rightarrow E\lbrack E\rbrack=1*\frac{1}{32}+8*P(2^3)+32P(2^5)+64P(2^6) \\ \Rightarrow E\lbrack E\rbrack=\frac{1}{32}+8*2^{-2}+32*2^{-4}+64*2^{-5} \\ \Rightarrow E\lbrack E\rbrack=\frac{1}{32}+8*\frac{1}{4}+32*\frac{1}{16}+64*\frac{1}{32} \end{gathered}[/tex]
Then,
[tex]\Rightarrow E\lbrack E\rbrack=\frac{1}{32}+2+2+2=6+\frac{1}{32}=\frac{193}{32}[/tex]
The answer is 193/32
