Ri 2025 Y6T2W4Mathfocus Summary+on+DRV Binomial

Page 1 of 3 RAFFLES INSTITUTION H2 Mathematics 9758 2024 Year 6 Term 1 Revision (Summary) Topic: Discrete Random Variable, Binomial Distribution Summary for Discrete Random Variable Probability Distribution A table or formula giving the values of P( )X x for every x in sample space S is called the probability distribution of X. For example, the experiment of tossing a fair coin 3 times where X is the number of heads obtained, the probability distribution of X is as follows: x 0 1 2 3 P( )X x 1 8 3 8 3 8 1 8 The probability distribution of X satisfies the following: 1. 0 P( ) 1X x   for all x in S. 2. P( ) 1 x S X x    where the summation is over all values of x in S. Expectation of Discrete Random Variable The expectation (or mean, or expected value) of a discrete random variable X taking values from a set S is given by E( ) P( ) x S X x X x    . In the example above,  1 3 3 1 3E 0 1 2 38 8 8 8 2X          Independent Discrete Random Variables Let X and Y be two discrete random variables taking on possible values 1 2, ,...x x and 1 2, ,...y y respectively. The random variables and are said to be independent if for all i and j, P( and ) P( )P( )i j i jX x Y y X x Y y     . In the example above, if T is the event that the outcome of tossing a coin is head, then             33 1 2 3 1 2 3 1 1P 0 P 0, 0, 0 P 0 P 0 P 0 P 0 2 8X T T T T T T T                      This is possible given that 1 2 3, andT T T are independent. X Y

Page 2 of 3 Functions of a Discrete Random Variable The expectation of g( )X , where g is a function of X, is denoted by E(g( )) g( )P( ). x S X x X x    In particular, 2 2E( ) P( ). x S X x X x    Variance and Standard Deviation of a Discrete Random Variable The variance of a discrete random variable X is given by       22 2 2 2 2 Var( ) E( ) P E P X X x X x X x X x                    . In the example above,   2 2 2 2 21 3 3 1 3 3Var 0 1 2 38 8 8 8 2 4X              The standard deviation of X, denoted by , is defined as Var( )X . Properties of Expectation and Variance (Note that these properties hold for discrete and continuous random variables) Let X and Y be random variables and a and b be constants. We have Expectation Variance (1) E( )a a (2) E( ) E( )aX a X (3) E( ) E( )aX b a X b   (4) E( ) E( ) E( )aX bY a X b Y   (1) Var( ) 0a  (2) 2Var( ) Var( )aX a X (3) 2Var( ) Var( )aX b a X  If X and Y are independent random variables, then (4) 2 2Var( ) Var( ) Var( )aX bY a X b Y  Important Results In general, E( ) E( )nX n X 1 2 1 2E( ... ) E E ... E En nX X X X X X n X        but 2Var( ) Var( )nX n X 1 2 1 2Var( ... ) Var( ) Var( ) ... Var Var( )n nX X X X X X n X        .

Page 3 of 3 Summary for Binomial Distribution Note that Binomial random variable is a special discrete random variable. Conditions for an experiment to follow binomial distribution 1. It consists of n (fixed number of repeated) independent trials. 2. The outcome of each trial is either a success or a failure. 3. The probability of success for each trial, denoted by p, remains constant. Note that the above must be stated in context of a given question. For example, a biased coin has probability 0.56 of obtaining a head in any toss. Find the probability of getting 6 heads if the coin is tossed 8 times. Conditions in context as follow 1. The coin is tossed independently 8 times. 2. The outcome of each toss is either a head (success) or tail (failure). 3. The probability of obtaining a head (success) remains constant at 0.56. Binomial Distribution If a discrete random variable X follows a binomial distribution, we write B ,X n p , where the parameters of the distribution n and p refer to the number of trials and the probability of success for each trial respectively. The probability distribution of X is given by (in MF27) P( ) (1 ) x n xnX x p px       , for 0,1 ,2,...,x n , where ! ( ! ! n x n nCx n x) x       The mean and variance of X are given by (i) E( )X np (in MF27) (ii) Var( ) (1 )X np p  (in MF27)