The probability distribution of a discrete random variable X is given by: P(X = −1) = 1 5 ,P(X = 0) = 2 5 and P(X = 1) = 2 5 (a) Compute E[X]. (b) Determine the probability distribution Y = X 2 and use it to compute E[Y ]. (c) Determine E £ X 2 ¤ using the change-of-variable formula. (You should match your answer in part (b). (d) Determine V ar (X).

Answer:Step-by-step explanation:Given that the probability distribution of a discrete random variable X is given by: [tex]P(X = −1) = \frac{1}{5} \\ P(X = 0) =\frac{2}{5}\\ P(X = 1) = \frac{2}{5}[/tex]a) [tex]E(x) =\Sigma x_i p_i = -1(\frac{1}{5} )+0((\frac{2}{5} )+1((\frac{2}{5} )\\=(\frac{1}{5} )[/tex]b) [tex]Y=x^2[/tex]So y can take values as 0 and 1.P(Y=0) = [tex]\frac{1}{5}[/tex]P(Y=1) = [tex]\frac{4}{5}[/tex](since -1 also becomes +1 when squared)E(Y) = [tex]0(\frac{1}{5})+1(\frac{4}{5})\\=\frac{4}{5}[/tex]c) [tex]E(x^2)=\Sigma x^2 p = \frac{4}{5}[/tex]d) Var(x) = [tex]\frac{4}{5} -\frac{1}{25} =\frac{19}{25}[/tex]