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Question

Find the expected value, variance and standard deviation of a random variable whose $$p.m.f$$ is.
$$X=x$$$$1$$$$2$$$$3$$
$$p(X=x)$$$$\frac{1}{5}$$$$\frac{2}{5}$$$$\frac{2}{5}$$


Solution

 $$X=x$$ 1
 $$P(X=x)$$ 0.2 0.4 0.4
$$\rightarrow $$ PMF (probability mass function)
Expected value $$=E[X]=0.2\times 1+0.4\times 2+0.4\times 3$$
                           $$=2.2$$
Variance $$=E[X^{2}]-(E[X])^{2}$$
$$E[X^{2}]=0.2\times 1^{2}+0.4\times 2^{2}+0.4\times 3^{2}=5.4$$
Variance $$=5.4-(2.2)^{2}=0.56$$
Standard deviation $$=\sqrt{variance}=\sqrt{0.56}=0.748$$

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