Sessions 1-20 15MT2005

# 27 x 130 101 27 p 059z107pz107 pz 059 08577

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=P(8.5≤X≤13.0)=P( 8.5 10.1 2.7 ≤ X ≤ 13.0 10.1 2.7 )=P(-0.59≤Z≤1.07)=P(Z≤1.07)-P(Z≤- 0.59) =0.8577-0.2776=0.5801. b) 0.5801 is the proportion of the product having maximum attenuation between 8.5 and 13.0 dB c) Proportion of the products having maximum attenuation greater than 15.1 dB =P(X>15.1)=P(Z>(15.1-10.1)/(2.7))=P(Z>1.85)=1-0.9678=0.0322 Problems to be discussed by the faculty: 1. Given a Standard Normal distribution, find the area under the curve which lies a) To the left of z=1.43; b) to the right of z=-0.89 c) between z=-2.16 and z=-0.65 d) to the left of z=-1.39 e) to the right of z=1.96 f) between z=-0.48 and z= -1.74

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111 2. Given a standard normal distribution find the value of k such that a) P ( z < k ) = 0.0427 ; b) P ( z > k ) = 0.2946 ; c) P(-0.93<Z<k)=0.7235.
112 Tutorial: 1.Given the normally distributed variable X with mean 18 and standard deviation 2.5, find a) P(X<15); b) the value of k such that P(X<k)=0.2236; c) the value of k such that P(X>k)=0.1814; d) P(17<X<21).

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113 Session: 12 Uniform distribution: Tutorial problems: 1. Find the distribution of a random variable having a uniform distribution on [0,1] 2. In a certain experiments, the error made in determining the solubility of a substance is a random variable having the uniform density with A=-0.025 and B=0.025. What are the probabilities that such an error will be a) Between 0.010 and 0.015 b) Between -0.012 and 0.012?
114 Homework: 1. From experience Mr. Harris has found that the low bid on a construction job can be regarded as a random variable having the uniform density f ( x ) = { 3 4 c ,for 2 c 3 < x < 2 c 0, elsewhere Where c is his own estimate of the cost of the job, what percentage should Mr. Harris add to his cost estimate when submitting bids to maximize his expected profit?

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115 MATLAB Commands: y=unifpdf(x,a,b) Computes the continues Uniform pdf at the given value of x using the corresponding lower end point (minimum) a and upper end point (maximum) b. Y=unifcdf(x,a,b) compute the cumulative distribution function of x using the corresponding lower end point and upper end point. With reference to the above Example- MATLAB Code: >> y=unifcdf(15,0,30)+unifcdf(30,0,30)-unifcdf(20,0,30); >>fprintf('\n probability that the passenger waits more than 5 minutes for a bus=%.4f\n',y) probability that the passenger waits more than 5 minutes for a bus=0.8333 Exponential distribution: Tutorial: 1. The life, in years, of a certain type of electrical switch has an exponential distribution with an average life β=2. If 100 of these switches are installed in different systems, what is the probability that at most 30 fail during the first year?
116 2. Suppose that a study of a certain computer system reveals that the response time, in seconds, has an exponential distribution with a mean of 3 seconds. a) What is the probability that response time exceeds 5 seconds? b) What is the probability that response time is less than 10 seconds? c) What are the mean and variance of response time? 3. The density function of the time Z in minutes between calls to an electrical supply store is given by f ( Z ) = { 1 10 e z 10 0, elsewhere , 0 < z < a) What are the mean and variance of time between calls?

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