Understanding Data For Data Analytics, Data Science, and Machine Learning – Part-4

Things to know beforehand

• sample space : In probability theory, the sample space (also called sample description space or possibility space) of an experiment or random trial is the set of all possible outcomes or results of that experiment.

[6] More about Kurtosis and Skewness

Kurtosis

There are 3 types of Kurtosis

• Leptokurtic : when the shape density of data is very high in middle, width is less and height is high.
• Mesokurtic : when the shape of date is symmetrical like normal distribution.
• Platykurtic : when data is spread out, width of distribution is high, and height is low.

If your kurtosis is highly positive, that may show that your sample is accurate because your central tendency represents your data well.

Note:
Not true for all the conditions.

Skewness

The skewness of the distribution is majored using the mean, median, and mode.
When the mean = median = mode the distribution is symmetrical, if your value of skewness shifts the distribution is no longer symmetrical.

• When the skewness is positive, a tail on the right of the distribution will be there.
• When the skewness is negative, a tail on the left of the distribution will be there.

[7] Introduction to Probability

Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes (“heads” and “tails”) are both equally probable; the probability of “heads” equals the probability of “tails”; and since no other outcomes are possible, the probability of either “heads” or “tails” is 1/2 (which could also be written as 0.5 or 50%).

$$Probability = \frac{n_{e}}{N}$$

• n, is an event you are interested in
• N, is total number of event

Basic Properties of Probability

• The probability of a sure event or certain event is 1.
• The probability of an impossible event is 0.
• The probability of an event E is a number P(E) such that 0 ≤ P (E) ≤ 1. Probability is always a positive number.
• If A and B are 2 events that are mutually exclusive, then P(A⋃B) = P(A) + P(B).
• An elementary event is an event having only one outcome. The sum of the probabilities of such events of an experiment is 1.
• The sum of probabilities of an event and its complementary event is 1. P(A) + P(A’) = 1.
• P(A⋃B) = P(A) + P(B) – P(A⋂B).
• P(A⋂B) = P(A) + P(B) – P(A⋃B).
• If A1, A2, A3, ………, And are mutually exclusive events, then P (A1 ⋃ A2 ⋃ A3… ⋃ An) = P(A1) + P(A2) + ………. + P (An)

Sets

Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole.

Sets are an easy topic to get started with, lean those
Resources:

1. https://www.khanacademy.org/math/in-in-grade-11-ncert/x79978c5cf3a8f108:sets
2. https://www.britannica.com/science/set-theory/Axiomatic-set-theory
3. https://plato.stanford.edu/entries/set-theory

Combinatorics

Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science.

Combination

Combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange, and a pear, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple and an orange; or a pear and an orange. More formally, a k-combination of a set S is a subset of k distinct elements of S. So, two combinations are identical if and only if each combination has the same members. (The arrangement of the members in each set does not matter.) If the set has n elements, the number of k-combinations, denoted as Cnk, is equal to the binomial coefficient.

$${\displaystyle {\binom {n}{k}}={\frac {n(n-1)\dotsb (n-k+1)}{k(k-1)\dotsb 1}},}{\displaystyle {\binom {n}{k}}={\frac {n(n-1)\dotsb (n-k+1)}{k(k-1)\dotsb 1}}}$$

Permutation

Permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word “permutation” also refers to the act or process of changing the linear order of an ordered set.

$${n} P{r}=\frac{n !}{(n-r) !}$$

Conditional Probability

Conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion, or evidence) has already occurred. This particular method relies on event B occurring with some sort of relationship with another event A. In this event, the event B can be analyzed by a conditionally probability with respect to A. If the event of interest is A and the event B is known or assumed to have occurred, “the conditional probability of A given B”, or “the probability of A under the condition B”, is usually written as P(A|B) or occasionally PB(A). This can also be understood as the fraction of probability B that intersects with

$$A:{\displaystyle P(A\mid B)={\frac {P(A\cap B)}{P(B)}}},{\displaystyle P(A\mid B)={\frac {P(A\cap B)}{P(B)}}}$$

Statically Independence

two events are independent if and only if:

$$P(A|B)= P(A)$$

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for part-5 go here