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SMA 2230 PROBABILITY AND STATISTICS II UNIVERSITY EXAMINATION 2013/2014 FIRST SEMESTER EXAMINATION FOR THE DEGREE OF BACHELOR OF SCIENCE (MATHEMATICS AND COMPUTER SCIENCE) SMA 2230: PROBABILITY AND STATISTICS II th Time: 9.00 – 11.00am Date: Monday 10 February 2014 INSTRUCTIONS ANSWER QUESTION ONE AND ANY OTHER TWO QUESTIONS Question One (30mks) cc) Distinguish between the following. (i) Discrete variables and continuous variables. (2mks) (ii) Binomial distribution and passon distribution (2mks) (iii) Type I Error and Type II Error (2mks) dd) The sample space of a random experiment is {a,b,c,d,e,f} and each outcome is equally likely. A random variable is defined as follows: Outcome a b 0 c 0 1.5 d e f 1.5 2 3 Determine i) The probability mass function of (4mks) ii) The mean of . (3mks) ee) A random variable X has the exponential distribution ( )={ i) − 0 0ℎ >0 Show that the moment generating function of is ( ) = (1 − )⁄ −1 (5mks) ii) Find the mean of . (4mks) ∽ N(0,1) and Pr( < ) = 0.7, Pr( > ) = 0.45 ff) Find Pr( < < ) (4mks) gg) A binomial distribution has mean 6 and standard deviation 2. Calculate , , . (4mks) Question Two (20mks) 2 +1 a) i) verify that ={ 25 = 0,1,2,3,4 0 ℎ is a probability mass function, and hence determine ii) Pr(2 ≤ (4mks) ≤ 4) (3mks) iii) Pr( > −10) (3mks) b) The probability density function of the time to failure of an electronic component in a − ⁄ 1000 computer is ( ) = { 1000 0 >0 ℎ Determine the probability that i) A component lasts more than 3000 hours. (3mks) ii) A component fails in the interval from 1000 to 2000 hours (4mks) iii) A component fails before 1000 hours (3mks) Question Three (20mks) a) A geometric random variable x has probability distribution. −1 i) ; = 1,2, … … ( ) = { (1 − ) 0 ℎ Show that the moment generating function is given by (( )= 1 − (1 − ) (5mks) ii) Use ( ) to find the mean and variance of . b) i) A random variable has a p.d.f (5mks) 2 0 Determine the probability density function of ( )={ 0< < 1 ℎ = 3 + 6 using the cumulative method technique. iii) (5mks) Suppose that parameter = 1 + 1 and 2 are two independent poisson random variables with 1 and 2 respectively. Find the probability density function of 2 using the moment generating function techniques. (5mks) Question Four (20mks) a) Let denote the number of bits received in Error in a digital communication channel, and assume that = 0.01. If 1000 bits are is a Binomial random variable with transmitted. Determine i) Pr( = 1) ii) Pr( 2) (2mks) iii) Mean and variance of (2mks) (2mks) b) (i) Suppose a random variable with moment generating function ( )= 3 +4 2 Determine the mean and variance of (4mks) (ii) The line width for a semi-conductor. Manufacturing is assumed to be normally distributed with a mean 0.5 m and standard duration of 0.05 m i) What is the probability that a line width is greater than 0.62 m . ii) What is the probability that a line width is between 0.47 iii) The line width of 90% of the sample is below what value? and 0.63 (6mks) c) Use Integration by parts to show that Γ( ) = ( − 1)Γ( − 1) (4mks) Question Five (20mks) a) Distinguish between Null hypothesis and Alternative hypothesis (4mks) b) A manufacturer is interested in the output voltage of a power supply used in a P.C. output, voltage is assumed to be normally distributed with standard duration 0.25 volts and manufacturer wishes to test. 0: Using i) / 1: ≠5 = 8 units The acceptance region is 4.85 ̅ 5.15 Find the value of ∝ ii) =5 (8mks) Find the power of the test for detecting a true mean output voltage of 5.17volts. (8mks)