Question

Im currently taking Stat I and is currently in need for some catching up.

Can someone please provide me the essential and important probability rules/ratios as well as for variance please. Thanks

Answer 1

Dear Student it is not a complete package since the area of probability is very wider and much complicated according to the situation. A detailed study is only possible through understanding the subject STATISTICS subject deeply. Here I provide you some important results in Probability. Hope you will be satisfy.

Mathematically, the probability is defined as the ratio of the number of favorable events to the total number of possible outcomes of a random experiment

IMPORTANT PROBABILITY RESULTS

• If two random variables say A, B are independent only when its Probability can be written as P (A and B) = P(A) * P(B)
• If A and B are any two events and are not disjoint, then P(A ∪ B) = P(A) + P(B) – P(A ∩ B). A and B are the subsets of sample space S. { Addition theorem of probability}
• If A and B are any two events and are disjoint, then P(A ∪ B) = P(A) + P(B). A and B are the subsets of sample space S.
• If A and B are any two events such that P(A) ≠ 0 and P(B) ≠ 0. If A and B are independent events then P(A ∩ B) = P(A). P(B). { multiplication theorem of probability}

Conditional Probability

P( A/B) = P(AB) / P(B)

P( B/A) = P(AB) / P(A)

P( A/B) = P(A) ; P(B/A) = P(B) { If A and B are independent }

Pairwise independence and Mutually independence

If A, B, C are three events then these three are pair wise independent only when P(AB)

=P(A)*P(B) and P(BC) = P(B)*P(C) and P(AC) = P(A)*P(C)

If they are Mutually independent they must be pairwise independent and P(ABC)=P(A)*P(B)*P(C)

We can extend this to any number of events.

coin / die : some probability results

If n fair coins are tossed, Total number of outcomes in the sample space = 2ⁿ ...The probability of getting exactly r-number of heads when n coins are tossed = _nC_r / 2ⁿ

If n dice are rolled then total number of outcomes = 6ⁿ  ... Each Equally likely outcome has Probability = 1/6ⁿ

You should look all permutation and combination formula and concept to understand some kinds of probability problems.

There theorems like Bayes theorem, Axiomatic approach of probability, Lot of inequalities, probability generating function etc.. For a starting student I provide some important results.. The later stage you can study all other details of probability...

Variance

Variance (σ²) in statistics is a measurement of the spread of data points.

Let V denotes variance; Some important results are:

V(x) = E (x²) - [E(x)]²   E denotes expectations and X denoted random variable

V(ax) = a² V(x) where a is constant

V (a+x) = V(x)

V(x+y) = V(x) + V(y) where x and y are independent random variable

V(x-y) = V(x) + V(y)

V(x+y) = V(x)+V(y)+2COV(x,y) where x and y are dependent random variables and COV is the covariance

V(x-y) = V(x)+V(y)-2COV(x,y)

There are some more results but it should need to understand conditional distribution , conditional expectation , pgf etc.. I think basically you need to study and understand the above results.. If you want more then comment me

Thank you DEAR STUDENT.....