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 $$ \begin{array}{lllll} \hline x & 20 & 30 & 40 & 50 \\ \hline p(x) & 0.6 & 0.2 & 0.1 & 0.1 \\ \hline \end{array} $$

Short answer

Calculate the weighted mean of the random variable by multiplying each outcome by its probability and summarizing the results. Repeat the same process for squared outcomes to find the expected value of the squares. Then, subtracting the square of the expected value from the expected value of the squares, will give you the variance. And, finally, the square root of the variance constitutes the standard deviation.

Step-by-step solution

Calculate the Mean of the Random Variable

The mean of a discrete random variable, often denoted \( \mu \) or \( E[X] \), is calculated as the sum of the product of each outcome and its probability. Using the given probability function, the mean can be calculated as: \( \mu = \sum x.p(x) = 20*0.6 + 30*0.2 + 40*0.1 + 50*0.1 \)

Compute the squares of the random variable values

This is needed to compute the variance as part of calculating the standard deviation. This can be done by squaring each value and multiplying by its corresponding probability: \( E[X^2] = \sum x^2.p(x) = 20^2*0.6 + 30^2*0.2 + 40^2*0.1 + 50^2*0.1 \)

Calculate the Variance of the Random Variable

The variance is the expectation of the squared deviation of a random variable from its mean: \( \sigma^2 = E[X^2] - (E[X])^2 \). Using the calculated value from Steps 1 and 2 in the formula, the variance can be computed.

Calculate the Standard Deviation of the Random Variable

The standard deviation is the square root of the variance: \( \sigma = \sqrt{\sigma^2} \). After finding the variance in step 3, take its square root to get the standard deviation.