NYJC 2026 FM Practice - FM Stats 1

NANYANG JUNIOR COLLEGE DEPARTMENT OF MATHEMATICS 2026 J2 H2 FM FM Statistics 1 - Discrete and Continuous Random Variable Instruction: Attempt the questions marked with *. 1* The piano shop opens at 9 am every day. The waiting time (in hours) between successive arrivals of customers at the shop has an exponential distribution with mean 1  . Records show that on average, the shop sees no customers in the first 5 minutes on 20% of the days. (a) Show that the mean number of customers in one hour is 12ln 5 . [2] (b) Given that the first customer has not arrived by 9.05 am, find the probability that the shop sees its first customer by 9.15 am. [2] (c) Find the probability that the 15th customer arrives after 10.30 am. [2] Records show that, on average, 1 in 10 of its customers bought a piano. Let W be the random variable denoting the number of customers served before the customer who bought the second piano. (d) Find P( ).Ww= [2] (e) By considering the mean and variance of an appropriate geometric distribution, find E( ).W [3] 2* The continuous random variable X has probability density function f given by 2 25 ( 1), 1 4,f ( ) 0, otherwise. xxx  + −  =  The random variable Y is defined by (a) Show that 4 25 P( ) for 0 1.Y y y y =   [3] (b) Find the cumulative distribution function of Y . [4] (c) Find the probability density function of Y. [2] 2.YX=

3 The number of distinct uranium deposits in a given area is a Poisson random variable with parameter 10. Independently, the number of distinct plutonium deposits in this area is a Poisson random variable with parameter 5. In a fixed period of time, each deposit is discovered independently with probability 1 50 . (a) Find the probability that no less than 2 deposits are discovered during this period. [2] (b) Given that no more than 3 deposits are found in this period of time, find the probability that there are at least 1 uranium deposit. [4] 4 A particle moves along a horizontal line, relative to a fixed origin, so that its position at a given time t is 0 sinx x t= , where 0x is a positive constant. Let X be the continuous random variable denoting the particle’s position. (a) Show that the cumulative density function of X is given by ( ) 0 1 00 0 0 1, 11F sin , 2 π 0, xx xx x x x x xx −   = + −      − and hence, find the probability density function of X. [5] (b) Find the exact probability that the magnitude of X is at most 1 02 x . [2] (c) Show that the expectation of X is 0, and find the exact variance of X using the substitution 0 sinxx = . [7] [H2 Math DRV] 5 Three fair six-sided die are thrown. The random variable X represents the highest of the three scores on the dice. (a) Show that 6P 9 1( 6) 1 2X == . [2] (b) By finding the probability distribution of X, show that E(X) = 119 24 and find the exact value of Var (X). [5]