For the random variable X with probability density function f(x) = \(\frac{(x-3)^2}{5}\) ; x = 3, 4, 5, the variance of X is:

The correct answer is  4/25.

Key Points

• The variance of a random variable X with probability density function f(x) is defined as the expected value of the squared deviation of X from its mean, and can be calculated as \(E[(X - E[X])^2].\)
• To find the variance of X in this case, you would need to know the full form of f(x) and calculate E[X] and \(E[(X - E[X])^2].\)
• Variance measures the spread or dispersion of a set of data or a random variable around its mean.
• A smaller variance indicates that the data is clustered closely around the mean, while a larger variance indicates that the data is more spread out.
• Variance is a important quantity in statistics and probability, and is used in many statistical models and hypothesis tests.
• Variance is always non-negative and its square root, the standard deviation, is a commonly used measure of spread.

 Additional Information

• Variance is a measure of the second moment or central moment about the mean of a set of data or a random variable.
• The variance of a random variable is also related to the expected value (mean) of the squared deviation of the random variable from its mean.
• The variance is an important quantity in the study of random variables, as it provides information about the distribution and spread of the data.
• In some cases, the variance may not provide a complete picture of the spread of the data. In such cases, other measures such as the coefficient of variation or interquartile range may be used.
• Variance is a key parameter in many statistical models, including regression analysis, analysis of variance (ANOVA), and others.
• Understanding and being able to calculate the variance is a fundamental skill for anyone working in the field of statistics and data analysis.