$ Sem5_maths Formulae

http://www.mube.co.nr Probability A 0 P(A) 1 ; P(S) =1 ; P(φ)= 0 ; P(A ) = 1- P( ) ≤ ≤ A B A B A B A B P( ) = P( ) ; P( ) = P( ) ∪ ∩ ∩ ∪ ....De Morgan’s Laws P(A B) = P(A) - P(A B) ; P(A ) = P(B) - P(A B) B ∩ ∩ ∩ ∩ P ( A B ) P ( A B ) P ( A / B)= P(B / A ) = P ( B ) P ( A) ; ∩ ∩ ....Conditional Probability P(B) = P(B/A 1 )*P(A 1 )+ P(B/A 2 )*P(A 2 )+ …..….. + P(B/A n )*P(A n ) ….Theorem of Total Pr obability j 1 1 2 2 n n ( ) * ( / ) P(A/B )= P( B/A )* P( A )+ P( B/A )* P( A )+ &...+P (B /A )* P( A ) P Aj P B Aj ....Bayes’ Theorem Probability Distribution A random variable X is rule that assigns a numerical value to each result in sample space of an experiment. A discrete random variable can take on specific, isolated numerical values, like the outcome of a roll of a die, or the number of dollars in a randomly chosen bank account. A continuous random variable can take on any values within a continuum or an interval (a, b), like the temperature in Central Park, or the height of an athlete in centimeters. Probability distribution of a discrete random variable is a list of probabilities associated with each of its possible values in interval (a, b). It is also called the probability function or the probability mass function (p.m.f) or probability density function (p.d.f). X x 1 x 2…… …..x n ( ) p xi 0 ≤ ≤ 1 ( ) p xi Σ =1 F(x i )=F(X= x i ) p(x 1 ) p(x 2 ) p(x n ) Cumulative distribution function (F) gives the probability that the random variable X is less than or equal to x, for every value x.It is also called distribution functio n or cumulative probability distribution. Cumulative distribution function for discrete variable is found by summing up the probabilities. For a continuous random variable, it is found by the integral of its probability densit y function. X x 1 x 2 x 3….. x n …… 0 ( ) 1 F X ≤ ≤ F(X)=0 for x<a F(X )=1 f or x>b F(x i )=F(X= x i ) p(x 1 ) 2 i 1 x i = ∑ 3 i 1 x i = ∑ i 1 x n i = ∑ ( ) ( ) ( ) x F X P X x f t d t = = = ∫ − ∞ .. for continuous random variable Probability Density Function f(x) The probability density function (p.d.f) of a continuous random variable is a function which can be integrated to obtain the probability that the random variable takes a value in a given interval. P.D.F, f(x) of a continuous random variable X is the derivative of the cumulative distribution function F(x). ( ) ( ) ( a ) P X f x d x b β α α β α β ≤ ≤ = ≤ ≤ ≤ ∫ Since F(x)=P(X≤x),it follows that, if f(x) is a probability density function then , I) Total probability for all possible values of the continuous random variable X is 1: II) Probability density function can never be negative: f(x) ≥ 0 for all x.

$ Sem5_maths Formulae

Documents

Transcript of $ Sem5_maths Formulae