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 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( , )Xn 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 MF26) P( ) (1 ) xn xnXx p p x    , for 0,1,2,...,x n , where ! (! ! n x n nCx nx ) x    The mean and variance of X are given by (i) E( )Xn p (in MF26) (ii) Var( ) (1 )X np p (in MF26)

Summary for Normal Distribution Note that Normal random variable is a special continuous random variable. For continuous random variable, probability is calculated using the area under its probability density function. In H2 Math syllabus, we just need to know how to use GC to evaluate these probabilities. We also need to be aware that  P P for discrete random variable (Binomial included)X<x X x BUT  P =P for continuous random variableX<x X x  Normal Distribution If a continuous random variable X follows a normal distribution, we write 2~N ( , )X  , where E( )X  and 2Var( )X  . Properties of a Normal Curve Let 2~N ( , )X  . (1) It is symmetri cal about the line x  . (2) The mean, median and mode are all equal to  . (3) It approaches the x axis as x . (4) Area under the graph gives the probabilities, i.e., P( ) f ( )d b a aXb xx  where f( )yx represents the probability density function of the normal curve. Hence P( )aXb is given by the area under the graph from xa to xb . (5) Total area under the curve is 1. b   a           

(6) P( ) 0.68 X    P( 2 2 ) 0.95X      P( 3 3 ) 0.997X     i.e., approximately 68%, 95% and 99.7% of the values drawn from a normal distribution lies within 1, 2 and 3 standard deviations of the mean respectively. Standard Normal Distribution Let 2~N ( , )X   . The random variable Z , which is the standard normal variable, is defined by XZ    . The standard normal distribution is ~N ( 0 , 1 )Z . The process of converting 2~N ( , )X   into ~N ( 0 , 1 )Z is known as standardization. P( )X x PP Xx x Z              Using the Properties of Expectation and Variance of Random Variables, we have the following results for independent Normal Random Variables Let 2 11~N ( , )X   and 2 22~N ( , )Y   be independent random variables and a and b be constants. We have (1) 22 12 1 2~N ( , )XY     (2) 22 11~N ( , )aX b a b a   (3) 22 2 2 121 2~N ( , )aX bY a b a b    

Revision Tutorial Questions Source of Question: ACJC Prelim 9758/2017/02/Q6 1. Alex and his friend stand ran domly in a queue with 3 other people. The random variable X is the number of people standing between Alex and his friend. (i) Show that P(X = 2) = 0.2. [2] (ii) Tabulate the probability distributi on of X. [2] (iii) Find E(X) and E(X – 1) 2. Hence find Var (X). [3] Solution 1 (i) 23 ! 1P( 2) P( ** *,* ** ) 2 0.2 (shown) 5! 5XA F A F     (ii) 23 ! 2P( 0) P( ***,* **,** *,*** ) 4 5! 5 0.4 23 ! 3P( 1) P( * **,* * *,** * ) 3 0.3 5! 10 P( XA F A F A F A F X A FA FA F           23 ! 13) P( *** ) 0.1 5! 10XA F      x 0 1 2 3 P(X = x) 0.4 0.3 0.2 0.1 (iii) E( ) P( ) 0(0.4) 1(0.3) 2(0.2) 3(0.1) 1 all x Xx X x      22 E( 1) ( 1) P( ) 1(0.4) 0(0.3) 1(0.2) 4(0.1) 1 all x Xx X x       22Var( ) E( ) E( 1) 1XX X  

Source of Question: JJC Prelim 9758/2017/02/Q5 2 The probability distribution of a discrete random variable, X, is shown below. x 1 2  P X x a b Find  E X and Var X in terms of a. [5] Solution: 2 1ba x 1 2  P X x a 1 a  E X =   12 1 aa = 2 a   2E X =     2212 1 aa = 43 a  Var X =    22EEXX  =  2 43 2aa =   243 44aa a = 2aa

Source of Question: MJC Prelim 9758/2017/02/Q6 3 The probability function of X is given by   21 i f 1 , 2 , 3 Pi f 4 0o t h e r w i s e xx Xx k x      where 10 9 . (i) Show that 19k  . Find, in terms of  , the probability distribution of X. [2] (ii) Find E X in terms of  and hence show that  2Var 26 196X   . [3] (iii) The random variable Y is related to X by the formula Ya b X , where a and b are non-zero constants. Given that  21Var 3Yb  , find the value of  . [3] Solution: Solution: (i) x 1 2 3 4  P X x  3 5