Mathematical Statistics

Concept:
The limit is a well-known fundamental limit in calculus. It defines the mathematical constant , which is the base of natural logarithms. Thus,
Step 1: Identify the standard limit form.
The expression given in the question exactly matches the standard definition of :
Step 2: Apply the known result.
Since the expression is the same as the definition of ,
Step 3: State the final result.

Concept:
A negatively skewed distribution (also called left-skewed distribution) has a longer tail on the left side. In such distributions, smaller values pull the mean toward the left. Because of this effect:

• The mean is pulled most toward the tail.
• The median lies between the mean and the mode.
• The mode remains near the peak of the distribution.

Thus, the relationship becomes:
Step 1: Understand the effect of skewness.
In a negatively skewed distribution, extreme low values pull the mean toward the left side.
Step 2: Determine the order of central tendencies.
Since the mean is affected the most by extreme values:
Step 3: State the conclusion.

Concept:
The integral is known as the Gaussian integral. It plays a fundamental role in probability theory, especially in the Normal distribution. A well-known result from calculus states: This result is usually derived by squaring the integral and converting it into polar coordinates.
Step 1: Define the integral.
Let
Step 2: Square the integral.

Step 3: Convert to polar coordinates.
Using and : Let , : Thus,
Step 4: Take the square root.

Concept:
The coefficient of determination is the square of the correlation coefficient. It represents the proportion of the variation in the dependent variable that is explained by the independent variable. If the correlation coefficient is , then: where is called the coefficient of determination.
Step 1: Identify the given correlation coefficient.

Step 2: Square the correlation coefficient.

Step 3: Interpret the result.
Thus, the coefficient of determination is:

Concept:
In Simple Random Sampling Without Replacement (SRSWOR), each unit in the population has an equal probability of being selected at any draw. Even though units are not replaced, the probability that any particular unit appears in a specific draw remains the same due to symmetry of selection. If the population size is , then the probability that a specific unit is selected in any particular position of the sample is:
Step 1: Consider the population size.
Let the population contain units, and suppose we are interested in the probability that a particular unit (say unit ) is selected at the second draw.
Step 2: Consider the two possible ways this can happen.
The specific unit can appear in the second draw if:

• It was not selected in the first draw.
• It is selected in the second draw.

Thus,
Step 3: Interpret the result.
Hence, the probability that a particular unit is selected at the second draw is:

Concept:
In statistical estimation, an estimator is said to be unbiased if its expected value equals the true population parameter . Mathematically, this is expressed as: This means that on average, the estimator correctly estimates the true parameter value.
Step 1: Understand the definition of an unbiased estimator.
If an estimator satisfies then the estimator does not systematically overestimate or underestimate the parameter.
Step 2: Identify the property.
The property that ensures the expected value of an estimator equals the population parameter is called Unbiasedness.