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Continuous Distributions BIC Prepaid By: Rajyagor Bhargav

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UNIFORM DISTRIBUTION BIC Prepaid By: Rajyagor Bhargav

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Characteristics and Applications: It is a continuous distribution. The uniform distribution is also referred to as rectangular distribution. The distribution has the constant height. Total Area of this distribution is 1. It deals with the problems like. “The time that a train takes to travel from Delhi to Agra. Suppose the train takes any time between 120 minutes and 150 minutes to reach Agra. Then it helps us in finding the probability that the train will take minutes between 135 to 140. BIC Prepaid By: Rajyagor Bhargav

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Contd…. Student complete Examination paper (Time). Average cost for insurance can vary in range. It helps us in finding probabilities when there is case of “Range within the Range” The variation of heights and weights of a students. Average income of managers in Pharma industries in Gujarat can vary in range. BIC Prepaid By: Rajyagor Bhargav

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Uniform Distribution Area = 1 a b BIC Prepaid By: Rajyagor Bhargav

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Example: Suppose a production line is set up to manufacture machine braces in lots of five per minute during a shift. When the lots are weighed, variation among the weights is detected, with lot weights ranging from 41 to 47 grams in a uniform distribution. Find the probability that a lot weighs between 42 and 45 grams. BIC Prepaid By: Rajyagor Bhargav

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Uniform Distribution of Lot Weights Area = 1 BIC Prepaid By: Rajyagor Bhargav

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Uniform Distribution Probability Area = 0.5 BIC Prepaid By: Rajyagor Bhargav

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Uniform Distribution Mean and Standard Deviation BIC Prepaid By: Rajyagor Bhargav

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Example: 6.1 Values uniformly distributed between 200 and 240. Find “f(x)”. Find mean and SD of this distribution. P(205<= x <= 220)? P(x>230)? P(x<=225)? BIC Prepaid By: Rajyagor Bhargav

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Example: 6.3 The retail price of a medium-sized box of a will known brand of cornflakes ranges from $2.80 to $3.14. Assume these prices are uniformly distributed. What is the average price and standard deviation of the prices? If price is randomly selected from the list, what is the probability that it will be between $3.00 and $3.10? BIC Prepaid By: Rajyagor Bhargav

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Example: 6.5 The average US household spends $2100 a tear on all types of insurance. Suppose the figures are uniformly distributed between the values of $400 and $3800. what is the height and SD of the distribution? What proportion of the households spends more than $3000 a year on insurance? More than $4000? Between $700 and $1500? BIC Prepaid By: Rajyagor Bhargav

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Characteristics of the Normal Distribution Continuous distribution Symmetrical distribution Asymptotic to the horizontal axis Unimodal A family of curves Area under the curve sums to 1. Area to right of mean is 1/2. Area to left of mean is 1/2. Mean, median and mode coincides for this distribution. 1/2 X BIC Prepaid By: Rajyagor Bhargav

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Applications: Almost everywhere for large n for the continuous type of data. Pharmaceutical research. Temperature from day to night in a standard condition. Spray of insecticides in a farm to prevent the crop. BIC Prepaid By: Rajyagor Bhargav

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Probability Density Function of the Normal Distribution X BIC Prepaid By: Rajyagor Bhargav

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Normal Curves for Different Means and Standard Deviations 2030405060708090100110120 BIC Prepaid By: Rajyagor Bhargav

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Standardized Normal Distribution A normal distribution with –a mean of zero, and –a standard deviation of one Z Formula –standardizes any normal distribution Z Score –computed by the Z Formula –the number of standard deviations which a value is away from the mean 1 BIC Prepaid By: Rajyagor Bhargav

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Z Table Second Decimal Place in Z Z0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.000.00000.00400.00800.01200.01600.01990.02390.02790.03190.0359 0.100.03980.04380.04780.05170.05570.05960.06360.06750.07140.0753 0.200.07930.08320.08710.09100.09480.09870.10260.10640.11030.1141 0.300.11790.12170.12550.12930.13310.13680.14060.14430.14800.1517 0.900.31590.31860.32120.32380.32640.32890.33150.33400.33650.3389 1.000.34130.34380.34610.34850.35080.35310.35540.35770.35990.3621 1.100.36430.36650.36860.37080.37290.37490.37700.37900.38100.3830 1.200.38490.38690.38880.39070.39250.39440.39620.39800.39970.4015 2.000.47720.47780.47830.47880.47930.47980.48030.48080.48120.4817 3.000.49870.49870.49870.49880.49880.49890.49890.49890.49900.4990 3.400.49970.49970.49970.49970.49970.49970.49970.49970.49970.4998 3.500.49980.49980.49980.49980.49980.49980.49980.49980.49980.4998 BIC Prepaid By: Rajyagor Bhargav

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-3-20123 Table Lookup of a Standard Normal Probability Z0.00 0.01 0.02 0.000.00000.00400.0080 0.100.03980.04380.0478 0.200.07930.08320.0871 1.000.34130.34380.3461 1.100.36430.36650.3686 1.200.38490.38690.3888 BIC Prepaid By: Rajyagor Bhargav

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Applying the Z Formula Z0.00 0.01 0.02 0.000.00000.00400.0080 0.100.03980.04380.0478 1.000.34130.34380.3461 1.100.36430.36650.3686 1.200.38490.38690.3888 BIC Prepaid By: Rajyagor Bhargav

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Normal Approximation of the Binomial Distribution The normal distribution can be used to approximate binomial probabilities Procedure –Convert binomial parameters to normal parameters –Does the interval lie between 0 and n? If so, continue; otherwise, do not use the normal approximation. –Correct for continuity –Solve the normal distribution problem±3 BIC Prepaid By: Rajyagor Bhargav

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Conversion equations Conversion example: Normal Approximation of Binomial: Parameter Conversion BIC Prepaid By: Rajyagor Bhargav

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Normal Approximation of Binomial: Interval Check 0102030405060 n 70 BIC Prepaid By: Rajyagor Bhargav

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Normal Approximation of Binomial: Correcting for Continuity Values Being Determined Correction XXXXXXXXXXXX +.50 -.50 +.05 -.50 and +.50 +.50 and -.50 BIC Prepaid By: Rajyagor Bhargav

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0 0.02 0.04 0.06 0.08 0.10 0.12 681012141618202224262830 Normal Approximation of Binomial: Graphs BIC Prepaid By: Rajyagor Bhargav

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Normal Approximation of Binomial: Computations 25 26 27 28 29 30 31 32 33 Total 0.0167 0.0096 0.0052 0.0026 0.0012 0.0005 0.0002 0.0001 0.0000 0.0361 XP(X) BIC Prepaid By: Rajyagor Bhargav

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Exponential Distribution Continuous Family of distributions Skewed to the right X varies from 0 to infinity Apex is always at X = 0 Steadily decreases as X gets larger Probability function BIC Prepaid By: Rajyagor Bhargav

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Graphs of Selected Exponential Distributions 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 012345678 BIC Prepaid By: Rajyagor Bhargav

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Exponential Distribution: Probability Computation 0.0 0.2 0.4 0.6 0.8 1.0 1.2 012345 BIC Prepaid By: Rajyagor Bhargav

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Example:6.27 Determine the following exponential probabilities. P(x>=5/ P(x<3/ P(x>4/ P(x<6/ BIC Prepaid By: Rajyagor Bhargav

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Example:6.29 A busy restaurant determined that between 6:30 p.m. and 9:00 p.m. on Friday nights, the arrival of customers are Poisson distributed with an average arrival rate of 2.44 per minute. a.What is the probability that at least 10 minutes will elapsed between arrivals? b.What is the probability that at least 5 minutes will elapsed between arrivals? c.What is the probability that at least 1 minute will elapsed between arrivals? d.What is the expected amount of time between arrivals? BIC Prepaid By: Rajyagor Bhargav

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Example: A manufacturing firm has been involved in statistical quality control for several years. As part of the production process, parts are randomly selected and tested. From the records ofs these tests, it has been established that a defective part occurs in a pattern that is Poisson distributed on the average of 1.38 defects every 20 minutes during production runs. Use this information to determine the probability that less than15 minutes will elapse between two defects. BIC Prepaid By: Rajyagor Bhargav

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