Question

The variance of a probability distribution is used to measure risk because a higher variance is associated with:_______

a. a wider spread of values around the mean.
b. a more compact distribution.
c. a lower expected value.
d. both a and b
e. all of the above

Answer 1

The variance of a probability distribution is associated with a wider spread of values around the mean, indicating more risk due to higher variability. The correct answer to the student's question is option (a). Variance and standard deviation are key concepts in understanding data spread and risk measurement.

Step-by-step explanation:

The variance of a probability distribution is used to measure risk because a higher variance is associated with a wider spread of values around the mean. Variance, symbolized as ², measures the degree of dispersion of a set of values from their mean (μ). When the variance is high, it indicates that the data values are spread out significantly from the mean, which can be interpreted as higher variability or risk in the context of a probability distribution. On the other hand, a low variance indicates data values that are closer to the mean, signifying less variability.

The formula for calculating the variance (²) of a discrete random variable X is ² = Σ (x − μ)² P(x), where x represents values of the random variable X, μ is the mean of X, P(x) represents the corresponding probability, and Σ indicates the sum of the squared differences multiplied by their probabilities. To obtain the standard deviation of a probability distribution, which is a measure of how far the outcomes of a statistical experiment are from the mean, we take the square root of the variance (²).

To answer the student's question, the correct option is (a) a wider spread of values around the mean. It is not associated with a more compact distribution or a lower expected value.

Answer 2

The appropriate solution is Option a (a wider spread of values around the mean).

Step-by-step explanation:

• Probability distributions could be characterized for random or unknown variables that are continuous and even some discrete.
• Intuition became linked to the number of random variables samples' attributes. The probability distribution would have any probability P(x) among zero (0) and one (1) for ordinal variables as well as the total of all of the other probabilities become one.

The many other suggestions made didn’t apply to the current case. Therefore the answer above would be correct.