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

$$X=x$$ 12 3  $$P(X=x)$$ 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 $$=\sqrt{variance}=\sqrt{0.56}=0.748$$Maths

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