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Question

A pair of fair dice is thrown. Let X be the random variable which denotes the minimum or equal of the two numbers which appear. Find the probability distribution, mean and variance of X.


Solution

Let $$X$$ denote the event of getting twice the number. Then, $$X$$ can take the values $$1,2,3,4,5$$ and $$6$$.
Thus, the probability distribution of $$X$$ is given by
$$x$$ $$P(X)$$ 
$$1$$ $$\dfrac{11}{36}$$ 
$$2$$ $$\dfrac{9}{36}$$ 
$$3$$ $$\dfrac{7}{36}$$ 
$$4$$ $$\dfrac{5}{36}$$ 
$$5$$ $$\dfrac{3}{36}$$ 
$$6$$ $$\dfrac{1}{36}$$ 
$$x_i$$ $$p_i$$ $$p_ix_i$$ $$p_ix_i^2$$ 
$$1$$ $$\dfrac{11}{36}$$ $$\dfrac{11}{36}$$ $$\dfrac{11}{36}$$
$$2$$ $$\dfrac{9}{36}$$ $$\dfrac{18}{36}$$ $$1$$ 
$$3$$ $$\dfrac{7}{36}$$ $$\dfrac{21}{36}$$ $$\dfrac{63}{36}$$ 
$$4$$ $$\dfrac{5}{36}$$ $$\dfrac{20}{36}$$ $$\dfrac{80}{36}$$ 
$$5$$ $$\dfrac{3}{36}$$ $$\dfrac{15}{36}$$ $$\dfrac{75}{36}$$ 
$$6$$ $$\dfrac{1}{36}$$ $$\dfrac{6}{36}$$ $$1$$ 
  $$\sum p_ix_i=\dfrac{91}{36}=2.5$$ $$\sum p_ix_i^2= 8.4$$
$$\Rightarrow$$  Mean $$=\sum p_ix_i=6.25$$
$$\Rightarrow$$  Variance $$=\sum p_ix_i^2-(Mean)^2$$
                      $$=8.4-6.25$$
                      $$=2.15$$

Maths

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