ICAR AIEEA PG 2024 - Statistical Sciences

When a single six-faced die is thrown, the number of possible outcomes is 6. When two independent unbiased dice are thrown simultaneously, the total number of outcomes in the sample space $S$ is given by the fundamental principle of counting:

The outcomes range from $(1,1), (1,2), \dots, (6,6)$.

By basic set theory, the intersection of two sets is a subset of their union:

By the monotonicity property of probability measures, if $A \subseteq B$, then $P(A) \le P(B)$.

In a histogram, the area of each rectangular bar must be strictly proportional to the frequency of that class:

When class intervals are unequal, plotting height proportional to frequency distorts the distribution. To maintain area proportionality, the height of the rectangle must be adjusted to Frequency Density:

Let the $n$ numbers be $x_1, x_2, \dots, x_n$.
The arithmetic mean is given by:

The total sum can be written as:

Given that the sum of the last $n-1$ numbers is $k$ (i.e. $\sum_{i=2}^n x_i = k$):

A fundamental algebraic property of the Arithmetic Mean states that the sum of squares of deviations (and hence the mean square deviation, $\frac{1}{N}\sum (x_i - A)^2$) is minimum when taken about the arithmetic mean (i.e. when $A = \bar{x}$).
Proof: Let $S = \sum (x_i - A)^2$. Differentiating with respect to $A$ and setting to zero yields $-2\sum (x_i - A) = 0 \implies A = \frac{\sum x_i}{N} = \bar{x}$.
Note: The sum of absolute deviations $\sum |x_i - A|$ is minimum when taken about the Median.