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Find the mean, variance and standard deviation of the number of tails in three tosses of a coin.
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Solution
Let X denote the number of tails in three tosses of a coin. Then, X can take the values 0, 1, 2 and 3.
Now,
\[P\left( X = 0 \right) = P\left( HHH \right) = \frac{1}{8}, P\left( X = 1 \right) = P\left( \text{ THH or HHT or HTH }\right) = \frac{3}{8}\]
\[P\left( X = 2 \right) = P\left( \text{ TTH or THT or HTT }\right) = \frac{3}{8}, P\left( X = 3 \right) = P\left( TTT \right) = \frac{1}{8}\]
Thus, the probability distribution of X is given by
x  P(X) 
0 
\[\frac{1}{8}\]

1 
\[\frac{3}{8}\]

2 
\[\frac{3}{8}\]

3 
\[\frac{1}{8}\]

Computation of mean and step deviation
x_{i}  p_{i}  p_{i}x_{i}  p_{i}x_{i}^{2} 
0 
\[\frac{1}{8}\]

0  0 
1 
\[\frac{3}{8}\]

\[\frac{3}{8}\]

\[\frac{3}{8}\]

2 
\[\frac{3}{8}\]

\[\frac{6}{8}\]

\[\frac{12}{8}\]

3 
\[\frac{1}{8}\]

\[\frac{3}{8}\]

\[\frac{9}{8}\]

`∑`p_{i}x_{i} =\[\frac{3}{2}\]

`∑` p_{i}x_{i}^{2}=3 
\[\text{ Mean} = \sum p_i x_i = \frac{3}{2}\]
\[\text{ Variance } = \sum p_i {x_i}2^{}_{}  \left( \text{ Mean } \right)^2 \]
\[ = 3  \left( \frac{3}{2} \right)^2 \]
\[ = \frac{3}{4}\]
\[\text{ Step Deviation} = \sqrt{\text{ Variance} }\]
\[ = \sqrt{\frac{3}{4}}\]
\[ = 0 . 87\]
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