Data Science and Machine Learning (Theory and Projects) A to Z - Expectations: Variance

The video tutorial explains the transformation of random variables and how to compute the expected value of a transformed variable. It covers the concept of moments, including the second moment, and their importance in data analysis. The tutorial also delves into variance, its calculation, and its role in statistics, particularly in the context of normal distribution. The video concludes with a brief overview of the law of large numbers and its application to variance.

What is the expected value of a transformed random variable if the transformation is a function of the original variable?

• It is always equal to the original variable.

• It is always zero.

• It is calculated using the transformed probability function.

• It is calculated using the original probability function.

In the example provided, what is the expected value of the transformed variable when X equals 2?

• 4

• 9

• 21

• 12

What is the primary reason the probability of a transformed variable remains the same as the original?

• The transformation function is linear.

• The transformation is reversible.

• The likelihood of the variable does not change.

• The original variable is continuous.

What is the second moment of a random variable?

• The standard deviation of the variable.

• The expected value of the square of the variable.

• The variance of the variable.

• The expected value of the variable.

How are moments related to the central limit theorem?

• They are not related.

• Moments help in proving the central limit theorem.

• Moments are only used in discrete variables.

• Moments are used to calculate probabilities.

What does variance measure in a dataset?

• The minimum value in the dataset.

• The maximum value in the dataset.

• The spread or variability of the dataset.

• The average value of the dataset.

How is variance calculated for a random variable?

• By adding all values and dividing by the number of values.

• By squaring the deviation from the mean and taking the expected value.

• By multiplying the mean by the standard deviation.

• By taking the square root of the mean.

What is the relationship between variance and the parameters of a normal distribution?

• Variance is always zero in a normal distribution.

• Variance is unrelated to the parameters.

• Variance is the square of the standard deviation.

• Variance is the mean of the distribution.

What does the law of large numbers support?

• The standard deviation.

• The sample variance.

• The median.

• The sample mean.

What happens to the probability model when a variable is transformed?

• It remains the same as the original model.

• It becomes undefined.

• It becomes a continuous model.

• It changes to a new model.