Question

In Exercises \(\mathrm{P} .78\) to \(\mathrm{P} .81,\) use the probability function given in the table to calculate: (a) The mean of the random variable (b) The standard deviation of the random variable $$ \begin{array}{lccc} \hline x & 10 & 20 & 30 \\ \hline p(x) & 0.7 & 0.2 & 0.1 \\ \hline \end{array} $$

Short answer

The mean of the random variable \(X\) is calculated as \(E(X)\) and the standard deviation as \(\sigma\).

Step-by-step solution

Calculate the Mean

To find the mean or expected value of the random variable, multiply each value of the random variable by its corresponding probability and then add up these products. Let's given the random variable \(x\) takes on the values 10, 20, and 30, with corresponding probabilities 0.7, 0.2, and 0.1 respectively. Then the expected value \(E(X)\) is computed as: \[E(X) = (10)(0.7) + (20)(0.2) + (30)(0.1)\]

Compute the Variance

To compute the variance of the random variable, subtract the mean from each random variable value, square the result, multiply it with the corresponding probability and sum all these calculated values. This is done as follows: \[Var(X) = [(10 - E(X))^2](0.7) + [(20 - E(X))^2](0.2) + [(30 - E(X))^2](0.1)\]

Calculate the Standard Deviation

The standard deviation is the square root of the variance. We calculate it as: \[\sigma = \sqrt{Var(X)}\]