Question

The probability density function for the random variable \(X\) defined to be the number of cars owned by a randomly selected family in Millinocket is given as $$\begin{array}{l|c|c|c|c|c}x & 0 & 1 & 2 & 3 & 4 \\\\\hline p(X=x) & 0.08 & 0.15 & 0.45 & 0.27 & 0.05\end{array}$$ Compute the variance and standard deviation of \(X\).

Short answer

Variance is approximately 0.9364 and standard deviation is about 0.9688.

Step-by-step solution

Understand the Formula for Variance

The variance of a random variable \(X\) is calculated using the formula: \( \text{Var}(X) = E(X^2) - [E(X)]^2 \). Here, \(E(X)\) is the expected value of the random variable \(X\), and \(E(X^2)\) is the expected value of \(X^2\).

Calculate the Expected Value \(E(X)\)

To find the expected value \(E(X)\), we use the formula: \(E(X) = \sum x_i \cdot p(X=x_i)\). Substitute the given values: \(E(X) = 0 \cdot 0.08 + 1 \cdot 0.15 + 2 \cdot 0.45 + 3 \cdot 0.27 + 4 \cdot 0.05 = 0 + 0.15 + 0.9 + 0.81 + 0.2 = 2.06\).

Calculate \(E(X^2)\)

To find \(E(X^2)\), use the formula: \(E(X^2) = \sum x_i^2 \cdot p(X=x_i)\). Substitute the given values: \(E(X^2) = 0^2 \cdot 0.08 + 1^2 \cdot 0.15 + 2^2 \cdot 0.45 + 3^2 \cdot 0.27 + 4^2 \cdot 0.05 = 0 + 0.15 + 1.8 + 2.43 + 0.8 = 5.18\).

Calculate Variance \(\text{Var}(X)\)

Insert \(E(X)\) and \(E(X^2)\) into the variance formula: \(\text{Var}(X) = 5.18 - (2.06)^2 = 5.18 - 4.2436 = 0.9364\). So, the variance is approximately 0.9364.

Calculate Standard Deviation

The standard deviation is the square root of the variance. So, take the square root of 0.9364: \(\text{SD}(X) = \sqrt{0.9364} \approx 0.9688\).

Key concepts

Probability Density Function

A probability density function (PDF) describes the likelihood of a continuous random variable taking on a range of values. In the context of our exercise, while the function provided is technically a probability mass function because it deals with a discrete random variable with specific values, the purpose remains the same—to define the probability of various outcomes.
A PDF is essentially a function that outlines how the probability is distributed across different values in a given dataset. For example, in our case, we have values from 0 to 4, each with a probability of occurrence. The PDF is helpful because:

• It allows quick determination of how likely an event is to occur.
• Facilitates computation of expected values, variances, and other statistical measures.

Understanding the PDF is crucial as it underpins many aspects of statistical analysis.

Expected Value

The expected value is a fundamental concept in statistics that represents the average outcome you would expect to see if you could repeat the random experiment many times. It is essentially a weighted average of all possible values that a random variable can take, each weighted by its probability.

For our random variable, the number of cars owned, the expected value calculation involves multiplying each number of cars by its respective probability, and then summing these products. This can be seen in our calculation: \(E(X) = 0 \cdot 0.08 + 1 \cdot 0.15 + 2 \cdot 0.45 + 3 \cdot 0.27 + 4 \cdot 0.05 = 2.06\).

• This value indicates that, on average, a randomly selected family in Millinocket owns about 2.06 cars.
• The expected value provides a simple summary of the dataset.

It's a key component of many statistical formulas and is used in calculating variance.

Standard Deviation

Standard deviation is a measure of the amount of variation or spread in a set of values. It gives insight into how much individual data points deviate from the mean or expected value.

In our exercise, after determining the variance, standard deviation is simply the square root of that variance. We calculated it as follows: \(\text{SD}(X) = \sqrt{0.9364} \approx 0.9688\).

• A lower standard deviation indicates that the data points are closer to the expected value.
• A higher standard deviation suggests greater variability.

This calculation is crucial for understanding the dispersion within a dataset, which in our example is quite moderate, suggesting that car ownership numbers do not drastically differ among families.

Random Variable

A random variable is a numerical description of the outcome of a statistical experiment. In simple terms, it is a variable that can take on different values, each with an associated probability, as a result of some random process.

In this exercise, the random variable is the number of cars owned by families. It can take values such as 0, 1, 2, 3, or 4, each with its probability defined by the probability function. Understanding random variables involves:

• Identifying all possible outcomes of the random process (in this case, car ownership numbers).
• Assigning a probability to each outcome.

Random variables are foundational in probability and statistics, as they enable the application of statistical measures, such as expected value and variance, to quantify and analyze real-word phenomena.