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Let X have the probability mass function P(X = −1) = 1 2 , P(X = 0) = 1 3 , P(X = 1) = 1 6 Calculate E(|X|) using the approaches in (a) and (b) below. (a) First find the probability mass function of the random variable Y = |X| and using that compute E(|X|). (b) Apply formula (3.24) with g(x) = |x|. For reference, formula 3.24 states

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Answer:

Step-by-step explanation:

From the given information:

Note that the possible values of Y are 0 and 1 because;

y = 0 if X = 0 and y = 1 if X = ±1

∴

[tex]P(Y =0) = P(X = 0) =\dfrac{1}{3}[/tex]

[tex]P(Y = 1) = P(X = -1 \ or \ 1) \\ \\ = P(X = -1) + P(X = 1)[/tex]

[tex]= \dfrac{1}{2}+ \dfrac{1}{6}[/tex]

[tex]=\dfrac{3+1}{6}[/tex]

[tex]= \dfrac{2}{3}[/tex]

b)

[tex]E(|X|) = \sum |x| P(X=x) = ( 1 \times \dfrac{1}{2}) + ( 0\times \dfrac{1}{3}) + ( 1 \times \dfrac{1}{6})[/tex]

[tex]= \dfrac{2}{3}[/tex]

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