Question

Expectation of a random variable denotes

• A. Mean
• B. Median
• C. Variance
• D. Standard deviation

Hint:

Always remember that the first raw moment about the origin (\(\mu'_1\)) of any probability distribution is mathematically defined as the expected value \(E[X]\), which is identical to the population mean.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The mathematical expectation (or expected value) of a random variable is a fundamental concept in probability theory. It represents the long-term average value of a random variable over a large number of independent experiments.
Key Formula or Approach:
For a discrete random variable \(X\) with a probability mass function \(P(X = x_i)\), the expected value \(E[X]\) is calculated as: \[ E[X] = \sum_{i} x_i P(X = x_i) \] For a continuous random variable \(X\) with a probability density function \(f(x)\), the expected value is: \[ E[X] = \int_{-\infty}^{\infty} x f(x) \, dx \]

Step 2: Detailed Explanation:
The expected value of a random variable is the probability-weighted average of all its possible values.
In physical mechanics, if we model the probability distribution as a mass distribution along a straight line, the expected value corresponds to the center of mass or center of gravity of that distribution.
In statistics, this center of mass is defined as the population mean (\(\mu\)).
Let us compare this with the other options:

Median: This is the middle value of a distribution, such that half of the observations are below it and half are above it. It represents the 50th percentile and does not necessarily equal the expected value unless the distribution is perfectly symmetric.

Variance: This is a measure of dispersion, describing how far the random variables are spread out from their expected value. It is defined mathematically as \(E[(X - E[X])^2]\).

Standard Deviation: This is the positive square root of the variance, expressing the dispersion in the same units as the random variable.
Therefore, expectation directly corresponds to the mean.

Step 3: Final Answer:
The expectation of a random variable denotes the Mean, which is Option (A).