Скачать или смотреть 12th Maths - Probability Distributions - Day 01 | CA KARAN KABRA

CA Karan Kabra
(CA, DISA, PGDIFM, CAIIB, M. Com, B. Com, DB&F, NET, SET)

All India Topper in Mathematics in CA (Scored 100%)
All India Topper in Statistics in CA (Scored 100%)
Winner of K.V. Chandramouli Memorial Prize “Best Paper Award” in Mathematics in CA
Winner of Ganeshmal Patni Memorial Prize “Best Paper Award” in Statistics in CA
All India Rank (AIR) 24th in CS Executive
1st in Maths in 12th Board (Scored 100%)

CA Karan Kabra has been in teaching field since 2013. In this span, he has taught over 8000 students the subject of Cost Accounting and Mathematics from 11th Std up to Chartered Accountancy Course.

The video transcript covers the foundational concepts of Probability Distributions, specifically focusing heavily on discrete random variables.
Here is the list of topics covered in the video:
1. Probability Distributions
◦ This is the main chapter topic, introduced as the eighth and final chapter of Part 2 of the course.
2. Random Variable
◦ Definition: A random variable is defined as a real-valued function defined on the sample space of a random experiment.
◦ Concept: It is used to define outcomes of an experiment, such as the number of heads when tossing coins.
◦ The collection of outcomes for a probability experiment, defined by a variable, is called a random variable.
3. Types of Random Variables
◦ There are two main types discussed.
4. Discrete Random Variable
◦ Definition: A random variable is discrete if its possible values form a countable set, which may be finite or infinite.
◦ Characteristics: The values are obtained by counting. The values cannot be decimals or fractions between integers (e.g., 1.5).
◦ Examples: Number of cars, number of pages, number of dogs, and number of stars (even though stars are infinite, they are still countable because counting can be started).
5. Continuous Random Variable
◦ Definition: A random variable is continuous if its possible values form an interval of real numbers.
◦ Characteristics: The values are obtained by measurement. It involves all possible decimals and fractions within a given interval. The possible values are not countable and can involve unending decimals.
◦ Examples: Weight, height, speed, temperature, and age, as these increase continuously and can take any value within a range.
6. Probability Distribution of a Discrete Random Variable
◦ Creation of a distribution table using the random variable X and its probability P.
◦ Example used: The probabilities associated with the number of heads (X=0,1,2,3) when three coins are tossed.
7. Probability Mass Function (PMF)
◦ The PMF is a concept specific to Discrete Random Variables.
◦ A probability distribution is called a PMF if it satisfies two criteria:
▪ Individual probabilities (P(X)) must be greater than or equal to zero (i.e., non-negative).
▪ The total summation of all probabilities (∑P(X)) must equal one.
◦ Calculation of probabilities using the PMF table, including finding the probability of X being less than, greater than, or equal to specific values.
8. Cumulative Distribution Function (CDF)
◦ Denoted by F(X).
◦ Concept: Similar to the 'less than type' cumulative frequency (CF).
◦ Calculation: It involves cumulatively adding up the probabilities (P) from the starting value up to the point in question.
◦ F(X=x) represents the cumulative value up to x, which is equivalent to P(X≤x).
9. Expected Value (E(X))
◦ Concept: Expected value is the mean or the center of gravity of the probability distribution. It is sometimes denoted by μ (mu).
◦ Formula: E(X)=∑Px. (This is analogous to ∑fx/∑f for frequency distributions, since ∑P=1).
◦ Meaning: It represents the statistical average outcome of the experiment (e.g., 1.5 heads expected from three coin tosses).
10. Expected Value of x^2 (E(x^2))
◦ Formula: E(x2)=∑Px2.
◦ The general principle is that the expected value of any function is found by multiplying that function by the probability P and taking the summation.
11. Variance of x (Variance(X) or \sigma^2)
◦ Formula 1 (using expected values): Variance(X)=E(x2)−E(x)2.
◦ Formula 2 (using summation): Variance(X)=∑Px2−E(x)2.
◦ Alternative Formula: Variance(X)=∑P⋅(x−E(x))2.
12. Standard Deviation of x (SD(X) or \sigma)
◦ Formula: Standard Deviation is the square root of the Variance.