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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).

Respuesta :

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]

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