RI Discrete Random Variable Binomial Distribution Normal Distribution Solns

RAFFLES INSTITUTION H2 Mathematics 9758 2023 Year 6 Term 3 Revision 14 (Summary and Tutorial) Topic: Discrete Random Variable, Binomial Distribution, Normal Distribution Summary for Discrete Random Variable Probability Distribution A table or formula giving the values of P( )Xx for every x in sample space is called the probability distribution of X. For 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( )Xx 1 8 3 8 3 8 1 8 The probability distribution of X satisfy the following: 1. 0P ( )1Xx  for all x in S. 2. P( ) 1 xS Xx   where the summation is over all values of x in S. Expectation of Discrete Random Variable The expectation (or mean, or expected va lue) of a discrete random variable X taking values from a set S is given by E( ) P( ) xS Xx X x   . Independent Discrete Random Variables Let X and Y be two discrete random variables taking on possible values 12, ,...xx and 12,, . . .yy respectively. The random variables and are said to be independent if for all i and j, P( and ) P( )P( )ij i jXx Yy Xx Yy    . 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( ). xS Xx X x   In particular, 22E( ) P( ). xS Xx X x   XY

Variance and Standard Deviation of a Discrete Random Variable The variance of a discrete random variable X is given by      22 22 2 2 Var( ) E( ) P EP XX x X x Xx X x              . 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( )aa (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) 22Var( ) Var( ) Var( )aX bY a X b Y  Important Results In general, E( ) E( )nX n X        12 1 2E( ... ) E E ... E EnnXX X X X X nX       but 2Var( ) Var( )nX n X  12 1 2Var( ... ) Var( ) Var( ) ... Var Var( )nnX XX X X X n X       .

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 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 toss