Probability Distribution
Once the data collected it needs to be sorted and analyzed. When the size of the data increases, analyzing them manually becomes impossible. It is then, the statistical tools comes to help. Statistical tools not only help in computing huge data but also its graphical representation helps in better representation of data.

Basic Statistics:

The catalyst for better analysis is sound statistics. So Statistics is logical and systematical way of studying the data, organizing them, making analysis of the data, interpret or decode them and finally presenting the data for final use. For such an elaborate process various methods are used for various purposes.

Following are the ways of studying, computing and presenting statistics:

• Probability
• Random variables
• Probability distribution
• The Central Limit Theorem
• Sampling and statistical inference
• Confidence intervals
• Hypothesis testing
PROBABILITY
It is the numerical way of depicting how probable an event is likely to offer. The process is often determined using set theory. The purpose is to analyze past data to find a pattern, assuming that past is a reflection of future.

• Set operations
Union (A U B)
Intersection (A B)

• Venn diagrams
Basic operations on Venn diagrams

• Basic probability axioms
P (S) = 1
P (A) >= 0 for all A S
P (A U B) = P(A) + P(B) –P (A B)

• Conditional probability
P(A|B) = P (A B)/ P(B)
• Bayes theorem
Random Variables
It’s a function or a rule which maps each event in a sample space to real numbers. For example:

Question: Suppose there are 8 balls in a bag. The random variable X is the weight, in lb, of a ball selected at random. Balls 1, 2 and 3 weigh 0.1lb, balls 4 and 5 weigh 0.15lb and balls 6, 7 and 8 weigh 0.2lb. Using the notation above, write down this information.

Solution: X(b1) = 0.10 lb, X(b2) = 0.10 lb, X(b3) = 0.1 lb,
X(b4) = 0.15 lb, X(b5) = 0.15 lb
X(b6) = 0.2 lb, X(b7) = 0.2 lb

There are two types of Random Variables
• Discrete Random Variables: The set of all possible values of the outcome. For Example:
Credit cards owned by an individual = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

• Continuous Random Variables: The set of possible values taken by a continuous random variable falls in an interval. For Example: Salary of a set of individuals
Discrete Probability Distributions
Uniform: The discrete uniform distribution is a symmetric probability distribution whereby a finite number of values are equally likely to be observed. Example: Assigning equal probability of default to a portfolio of credit card holders

Bernoulli: A Bernoulli trial is an experiment which has only two possible outcomes S (success) and f (failure). Examples: Tossing of a coin. "Head" corresponds to "success" and "Tail" corresponds to "failure"

Binomial: The probability of getting exactly " K " successes in " N " trials is given by the probability mass function.
Continuous Probability Distributions
Uniform: Assigns equal probability to all values between its minimum and maximum values. Example: Assigning equal probability of default to a portfolio of credit card holders.

Normal: A symmetrical distribution having bell shaped pdf curve which is widely used to naturally occurring variables e.g. height, weight, exam scores etc

Lognormal: A positively skewed distribution. Used the predict claim amount in Auto insurance

T-distribution: It is used when the sample size is small and the population standard deviation is unkown.

F- Distribution: It’s an asymmetric distribution that has 0 minimum value but no maximum value.

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