ACJC 2026 DRV and Binomial Distributions Summary

Anglo-Chinese Junior College 2025 H2 Mathematics 9758: DRV (with Binomial Distribution) / Summary / Page 1 of 2 SUMMARY: Discrete Random Variables (with Binomial Distribution) Approach to answering “DRV” Questions Example: For a game, three dice are thrown. If ‘1’ or ‘6’ appears you will be paid $1, but if neither ‘1’ nor ‘6’ appears, you will pay $5. Find the probability distribution of the amount that you can earn in 1 game. How much would you expect to lose in nine of these games? Solution: Let the random variable X denote the amount earned in one game. ( ) ( ) 8 19 7E P 5 1 27 27 9X x x X= = =−  +  =− ( ) ( )1 2 9E ... E 9 7X X X X+ + + =  =− Since the expectation is negative, I am expected to lose $7 in nine games. Expectation and Variance Expectation (or mean or expected value) of X (usually denoted by ) )P()E( all xXxX x == If g(X) is any function of the discrete random variable X, then   all E(g( )) g( ) P( ) x X x X x== . Variance of X (usually denoted by 2 ) ( ) 22 all Var( ) E ( ) P( ) x X X x X x = − = − =  We usually use ( )   22Var( ) E E( )X X X=− . Standard deviation of X (usually denoted by ) It is the positive root of the variance of X. ( )Var X = If a, b, and 12, , ..., na a a are real constants, and X and Y denote independent discrete random variables (with 12, , ..., nX X X denoting repeated independent observations of X), then (a) E( )aa = (b) E( )aX = E( )aX (c) E( ) E( )aX b a X b =  (d) ( )E E( ) E( )aX bY a X b Y =  (e) ( )11E ... nna X a X ( )1 ... E( )na a X=   (a) Var( ) 0a = (b) 2Var( ) Var( )aX a X= (c) ( ) 2Var Var( )aX b a X= (d) ( ) 22Var Var( ) Var( )aX bY a X b Y = + (e) ( )11Var ... nna X a X ( ) 22 1 .... Var( )na a X= + + x −5 1 P( )Xx= 4 4 4 8 6 6 6 27     =       8 191 27 27−= Step 1: Define the random variable. Step 2: Identify possible outcomes. Step 3: Calculate probability of each outcome. Step 4: Use probability distribution to answer subsequent parts.

Anglo-Chinese Junior College 2025 H2 Mathematics 9758: DRV (with Binomial Distribution) / Summary / Page 2 of 2 Binomial Distribution If the discrete random variable X represents the number of successes in n trials of an experiment, with each trial being a success with probability p, then we write ( )~ B ,X n p . ( )E X np= , ( ) ( )Var 1X np p=− (in MF27). The probability that there are r number of successes in the n trials is given by ( ) ( )P1 nrrnX r p p r −= = − (in MF27). Conditions under which a binomial distribution can be a suitable model: (1) The experiment consists of n independent repeated trials. (2) The result of each trial is only one of two possible outcomes , labe lled “success” or “failure”. (3) The probability of “success” of each trial remains constant and is denoted by p. If the question asks for assumptions, you should answer using points (1) and (3) above, in context. Example of answering in context: According to the Health Promotion Board, 28% of Primary 1 students in Singapore are short - sighted. A sample of 20 Primary 1 students are randomly selected for a survey. Explain why a binomial distribution is appropriate to model the number students in the sample who are short-sighted. Solution: A binomial distribution is appropriate because • there are only two possible outcomes: either a student is short-sighted or he is not. (2) • a randomly chosen student being short -sighted is independent of any other student being short-sighted, since the whole cohort of Primary 1 students in Singapore is large. (1) • the probability of a student being short-sighted is constant at 0.28. (3) Usage of GC: • To find probabilities: (1) Key in binompdf for ( )P Xr= . (2) Key in binomcdf for ( )P Xr . • To find unknowns: Key in binompdf or binomcdf into [y=] and use table or graph to find the unknown: Given In G.C, use Graph/Table 1~ B( , ),20Xn ( )P 10 0.995X  1binomcdf , ,1020YX =  Use ‘table’ (since n is a positive integer) ~ B(10, ),Xp ( )P 3 0.2013X == ( )binompdf 10, ,3YX= Use ‘graph’ - set window, Xmin, Xmax, Ymin, Ymax to between 0 and 1 (since p is a probability) ~ B(18,0.1),X ( )P 0.9Xr ( )binomcdf 18,0.1, 1YX=− Use ‘table’ (since r is a positive integer) Example of dealing with inequalities: If X is a discrete random variable, ( ) ( ) ( ) ( ) P 2 9 P 3 8 P 8 P 2 XX XX   =   =  − 