3.3.4 The Variance of X

• Definition of Expected Value:
  • The expected value $E[X]$ represents the center of the probability distribution.
• Physical Analogy:
  • Visualize placing point mass $p(x)$ at each value $x$ along a one-dimensional axis.
  • The axis is supported by a fulcrum positioned at $\mu$ (the expected value).
• Equilibrium Condition:
  • When the fulcrum is at $\mu$, the axis remains balanced, indicating no tendency to tilt.
  • This balance illustrates how $\mu$ serves as the point around which the distribution’s mass is centered.
• Implications of the Analogy:
  • The expected value provides insight into the overall behavior of the distribution.
  • It highlights the concept of central tendency in probability and statistics.
• Visual Representation:
  • Figure 3.7 (not included here) likely illustrates this analogy with two different distributions, demonstrating the concept of balance at the expected value.

Figure 3.7 Two different probability distributions with $\mu =4$

• Comparison of Distributions:

  • Both distributions in Figure 3.7 have the same expected value $\mu$.
  • However, the distribution shown in Figure 3.7(b) exhibits greater spread (variability or dispersion) compared to Figure 3.7(a).
  • Role of Variance:
    • To quantify the amount of variability in the distribution of $X$, we utilize the variance.
    • Variance, denoted as $\text{Var}(X)$ or ${\sigma }^2$, measures how much the values of $X$ deviate from the expected value $\mu$.
  • Relation to Sample Variability:
    • Just as $s^2$ (sample variance) was used in Chapter 1 to assess variability within a sample, the variance $\text{Var}(X)$ serves a similar purpose for a probability distribution.

• Implications of Variance:

  • A higher variance indicates a wider spread of values, reflecting greater uncertainty in predictions based on the distribution.
  • Understanding variance is essential for analyzing the reliability and variability of data.

• Conclusion:

  • Variance provides a crucial measure of dispersion, allowing for better insights into the behavior of random variables and their distributions.

variance

• Let $X$ have $\mathrm{pmf}p\left(x\right)$ and expected value $\mu$ .
• Then the variance of $X$ , denoted by $V\left(X\right)$ or > ${\sigma }_X^2$ , or just ${\sigma }^2$ , is

$$\begin{array}{rl}V\left(X\right)& =\sum _D{\left(x-\mu \right)}^2\cdot p\left(x\right)\\ & =E\left[{\left(X-\mu \right)}^2\right]\end{array}$$

The standard deviation (SD) of $X$ is ${\sigma }_X=\sqrt{{\sigma }_X^2}$

Squared Deviations and Variance:

• Definition of Squared Deviation:
  • The quantity $h(X)=(X-\mu {)}^2$ represents the squared deviation of $X$ from its mean $\mu$.
• Variance:
  • The variance ${\sigma }^2$ is the expected squared deviation, calculated as the weighted average of squared deviations.
  • Weights are determined by the probabilities from the distribution.
• Impact of Probability Distribution:
  • If most of the probability distribution is concentrated near $\mu$:
    • ${\sigma }^2$ will be relatively small.
  • If there are significant values of $x$ far from $\mu$ with large probabilities $p(x)$:
    • ${\sigma }^2$ will be quite large.
• Interpretation of Standard Deviation:
  • The standard deviation $\sigma$ can be viewed as a measure of the typical size of a deviation from the mean $\mu$.
• For example, if $\sigma =10$:
  • In a long sequence of observed $X$ values, some will deviate from $\mu$ by more than 10, while others will be closer.
  • The typical deviation from the mean will be on the order of 10.
• Conclusion:
  • Understanding squared deviations, variance, and standard deviation is crucial for assessing variability and interpreting the behavior of random variables in statistical analysis.

EX 3.24 check out DVD

When the pmf $p\left(x\right)$ specifies a mathematical model for the distribution of population values, both ${\sigma }^2$ and $\sigma$ measure the spread of values in the population;

• ${\sigma }^2$ is the population variance,
• $\sigma$ is the population standard deviation.