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The expected value is a fundamental concept in probability and statistics. It's the average or mean value that we expect to see from a random variable over a large number of trials. In simpler terms, it's the long-term average of outcomes. To calculate the expected value, we multiply each possible outcome of our random variable by its probability and sum up these products. This gives us a single number that represents the center of the distribution of the variable. In the context of investment optimism, the expected value helps us understand the average sentiment among investment managers. In our exercise, each sentiment category (very bullish, bullish, etc.) is assigned a numerical value from 5 down to 1. The expected value is calculated by:

• Multiplying each sentiment level by its probability:
• 5 x 0.04, 4 x 0.39, 3 x 0.29, 2 x 0.21, and 1 x 0.07
• Then summing these up to get 3.12.

This result shows that on average, investment managers hold an outlook just above neutral. It reflects the proportion of managers in each category and gives us a benchmark average sentiment.

Variance measures how much the values of a random variable deviate from the mean (or expected value). It tells us how spread out the outcomes are around the expected value. A larger variance means more spread, while a smaller variance indicates that outcomes are clustered closely around the mean. The calculation of variance involves:

• Finding the differences between each outcome and the expected value (mean).
• Squaring these differences to remove negatives and emphasize larger differences.
• Multiplying each squared difference by its probability.
• Summing these products to get the total variance.

In our example:

• The variance is computed as \((5-3.12)^2(0.04) + (4-3.12)^2(0.39) + \ldots\)
• This results in a variance of around 1.0248.

The small variance suggests that most manager outlooks are fairly close to the average sentiment. This indicates consistency in their views, with few extreme opinions.

Standard deviation is a statistic that measures the dispersion of a dataset relative to its mean, and it is calculated as the square root of the variance. While variance gives us a squared value, the standard deviation reverts this back to the original unit of the data. This makes it a more intuitive measure of variability because it is on the same scale as the data. If the standard deviation is small, the data points tend to be close to the mean. Conversely, a larger standard deviation signifies a wider range of values. In our problem, the standard deviation is calculated by taking the square root of the variance (1.0248), which is approximately 1.01. This tells us that the level of optimism among investment managers is not extremely varied, yet there is enough variability to indicate that there are different views on the market. The standard deviation provides insight into how much individual sentiment levels deviate from the expected value, giving us a clearer picture of overall sentiment.