- Calculate the expected value: \(E[X] = 1 \cdot \frac{1}{4} + 3 \cdot \frac{1}{4} + 4 \cdot \frac{1}{2} = 3\).
- Calculate the expectation of the squared values: \(E[X^2] = 1^2 \cdot \frac{1}{4} + 3^2 \cdot \frac{1}{4} + 4^2 \cdot \frac{1}{2} = 10.5\).
- Determine the variance: \(\text{Var}(X) = E[X^2] - (E[X])^2 = 10.5 - 3^2 = 1.5\).
- Find the standard deviation: \(\sigma = \sqrt{\text{Var}(X)} = \sqrt{1.5} \approx \boxed{1.22}\).
Explanation
- Analysis of the probability mass function
We are given the probability mass function (PMF) for a discrete random variable \(X\):
- For \(x=1\), \(P(X=1) = \frac{1}{4}\).
- For \(x=3\), \(P(X=3) = \frac{1}{4}\).
- For \(x=4\), \(P(X=4) = \frac{1}{2}\).
The objective is to find the standard deviation, \(\sigma_X\), of this random variable. The standard deviation is defined as the square root of the variance, \(\text{Var}(X)\). The variance can be calculated using the formula:
\[\text{Var}(X) = E[X^2] - (E[X])^2\]
where \(E[X]\) is the expected value of \(X\) and \(E[X^2]\) is the expected value of \(X^2\).
- Calculating the expected value \(E[X]\)
First, we find the expected value \(E[X]\), which is calculated by summing the products of each value \(x\) and its corresponding probability \(P(X=x)\):
\[E[X] = \sum x_i P(X=x_i) = (1 \cdot \frac{1}{4}) + (3 \cdot \frac{1}{4}) + (4 \cdot \frac{1}{2})\]
\[E[X] = \frac{1}{4} + \frac{3}{4} + \frac{4}{2} = \frac{4}{4} + 2 = 1 + 2 = 3\]
So, the expected value of \(X\) is \(3\).
- Calculating the expected value of the square \(E[X^2]\)
Next, we calculate the expected value of the square, \(E[X^2]\), by summing the product of each squared value \(x^2\) and its probability:
\[E[X^2] = \sum x_i^2 P(X=x_i) = (1^2 \cdot \frac{1}{4}) + (3^2 \cdot \frac{1}{4}) + (4^2 \cdot \frac{1}{2})\]
\[E[X^2] = (1 \cdot \frac{1}{4}) + (9 \cdot \frac{1}{4}) + (16 \cdot \frac{1}{2})\]
\[E[X^2] = \frac{1}{4} + \frac{9}{4} + \frac{16}{2} = \frac{10}{4} + 8 = 2.5 + 8 = 10.5\]
So, \(E[X^2] = 10.5\).
- Calculating the variance and standard deviation
Now we can find the variance using the formula \(\text{Var}(X) = E[X^2] - (E[X])^2\):
\[\text{Var}(X) = 10.5 - (3)^2\]
\[\text{Var}(X) = 10.5 - 9 = 1.5\]
Finally, the standard deviation is the square root of the variance:
\[\sigma_X = \sqrt{\text{Var}(X)} = \sqrt{1.5} \approx 1.2247\]
Rounding this value to the nearest option provided, we get \(1.22\).
Examples
Standard deviation is a vital tool in finance, especially when assessing investment risk. For example, if you compare two different stock portfolios with the same expected return but different standard deviations, the one with the lower standard deviation is typically considered less risky because its returns are more stable over time. By calculating the standard deviation, you quantify the volatility or "spread" of possible outcomes relative to the average.