Continous Random Variable.

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Name                                       Shakeel Nouman Religion                                  Christian Domicile                            Punjab (Lahore) Contact #             0332-4462527. 0321-9898767 E.Mail                                [email protected] [email protected]

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Continuous Random Variables:

Shakeel Nouman M.Phil Statistics Continuous Random Variables Continuous Random Variables By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Continuous Random Variables:

Continuous Random Variables 5.1 Continuous Probability Distributions 5.2 The Uniform Distribution 5.3 The Normal Probability Distribution 5.4 Approximating the Binomial Distribution by Using the Normal Distribution ( Optional ) 5.5 The Exponential Distribution ( Optional ) 5.6 The Cumulative Normal Table ( Optional ) Continuous Random Variables By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Continuous Probability Distributions:

Continuous Probability Distributions Recall: A c ontinuous random variable may assume any numerical value in one or more intervals Use a continuous probability distribution to assign probabilities to intervals of values The curve f( x ) is the continuous probability distribution of the continuous random variable x if the probability that x will be in a specified interval of numbers is the area under the curve f( x ) corresponding to the interval Other names for a continuous probability distribution: probability curve , or probability density function Continuous Random Variables By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Properties of Continuous Probability Distributions:

Properties of Continuous Probability Distributions Properties of f( x ): f( x ) is a continuous function such that 1. f ( x )  0 for all x 2. The total area under the curve of f( x ) is equal to 1 Essential point: An area under a continuous probability distribution is a probability Continuous Random Variables By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Area and Probability:

Area and Probability The blue-colored area under the probability curve f( x ) from the value x = a to x = b is the probability that x could take any value in the range a to b Symbolized as P ( a  x  b ) Or as P ( a < x < b ), because each of the interval endpoints has a probability of 0 Continuous Random Variables By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Distribution Shapes:

Distribution Shapes Symmetrical and rectangular The uniform distribution Section 5.2 Symmetrical and bell-shaped The normal distribution Section 5.3 Skewed Skewed either left or right Section 5.5 for the right-skewed exponential distribution Continuous Random Variables By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

The Uniform Distribution:

The Uniform Distribution If c and d are numbers on the real line ( c < d ), the probability curve describing the uniform distribution is The probability that x is any value between the given values a and b ( a < b ) is Note: The number ordering is c < a < b < d Continuous Random Variables By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

The Uniform Distribution Continued:

The Uniform Distribution Continued The mean m X and standard deviation s X of a uniform random variable x are These are the parameters of the uniform distribution with endpoints c and d ( c < d ) Continuous Random Variables By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Notes on the Uniform Distribution:

Notes on the Uniform Distribution The uniform distribution is symmetrical Symmetrical about its center m X m X is the median The uniform distribution is rectangular For endpoints c and d ( c < d ) the width of the distribution is d – c and the height is 1/( d – c ) The area under the entire uniform distribution is 1 Because width  height = ( d – c )  [1/( d – c )] = 1 So P ( c  x  d ) = 1 See panel a of Figure 5.2 Continuous Random Variables By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

The Normal Probability Distribution:

The Normal Probability Distribution The normal probability distribution is defined by the equation for all values x on the real number line, where is the mean and  is the standard deviation,  = 3.14159 … and e = 2.71828 is the base of natural logarithms Continuous Random Variables By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

The Normal Probability Distribution Continued:

The normal curve is symmetrical and bell-shaped The normal is symmetrical about its mean m The mean is in the middle under the curve So m is also the median The normal is tallest over its mean m The area under the entire normal curve is 1 The area under either half of the curve is 0.5 The Normal Probability Distribution Continued Continuous Random Variables By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Properties of the Normal Distribution:

Properties of the Normal Distribution There is an infinite number of possible normal curves The particular shape of any individual normal depends on its specific mean m and standard deviation s The highest point of the curve is located over the mean mean = median = mode All the measures of central tendency equal each other This is the only probability distribution for which this is true Continuous Random Variables By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Properties of the Normal Distribution Continued:

Properties of the Normal Distribution Continued The curve is symmetrical about its mean The left and right halves of the curve are mirror images of each other The tails of the normal extend to infinity in both directions The tails get closer to the horizontal axis but never touch it The area under the normal curve to the right of the mean equals the area under the normal to the left of the mean The area under each half is 0.5 Continuous Random Variables By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

The Position and Shape of the Normal Curve:

The Position and Shape of the Normal Curve The mean m positions the peak of the normal curve over the real axis The variance s 2 measures the width or spread of the normal curve Continuous Random Variables By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Normal Probabilities:

Normal Probabilities Suppose x is a normally distributed random variable with mean m and standard deviation s The probability that x could take any value in the range between two given values a and b ( a < b ) is P ( a ≤ x ≤ b ) P ( a ≤ x ≤ b ) is the area colored in blue under the normal curve and between the values x = a and x = b Continuous Random Variables By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

The Standard Normal Distribution #1:

The Standard Normal Distribution #1 If x is normally distributed with mean  and standard deviation , then the random variable z is normally distributed with mean 0 and standard deviation 1; this normal is called the standard normal distribution Continuous Random Variables By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

The Standard Normal Distribution #2:

The Standard Normal Distribution #2 z measures the number of standard deviations that x is from the mean m The algebraic sign on z indicates on which side of m is x z is positive if x > m ( x is to the right of m on the number line) z is negative if x < m ( x is to the left of m on the number line) Continuous Random Variables By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

The Standard Normal Table #1:

The Standard Normal Table #1 The standard normal table is a table that lists the area under the standard normal curve to the right of the mean ( z = 0) up to the z value of interest See Table 5.1 Also see Table A.3 in Appendix A and the table on the back of the front cover This table is so important that it is repeated 3 times in the textbook! Always look at the accompanying figure for guidance on how to use the table Continuous Random Variables By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

The Standard Normal Table #2:

The Standard Normal Table #2 The values of z (accurate to the nearest tenth) in the table range from 0.00 to 3.09 in increments of 0.01 z accurate to tenths are listed in the far left column The hundredths digit of z is listed across the top of the table The areas under the normal curve to the right of the mean up to any value of z are given in the body of the table Continuous Random Variables By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

The Standard Normal Table Example:

The Standard Normal Table Example Find P (0 ≤ z ≤ 1) Find the area listed in the table corresponding to a z value of 1.00 Starting from the top of the far left column, go down to “1.0” Read across the row z = 1.0 until under the column headed by “.00” The area is in the cell that is the intersection of this row with this column As listed in the table, the area is 0.3413, so P (0 ≤ z ≤ 1) = 0.3413 Continuous Random Variables By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Calculating P(-2.53 ≤ z ≤ 2.53) #1:

Calculating P (-2.53 ≤ z ≤ 2.53) #1 First, find P (0 ≤ z ≤ 2.53) Go to the table of areas under the standard normal curve Go down left-most column for z = 2.5 Go across the row 2.5 to the column headed by .03 The area to the right of the mean up to a value of z = 2.53 is the value contained in the cell that is the intersection of the 2.5 row and the .03 column The table value for the area is 0.4943 Continued Continuous Random Variables By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Calculating P(-2.53 ≤ z ≤ 2.53) #2:

Calculating P (-2.53 ≤ z ≤ 2.53) #2 From last slide, P (0 ≤ z ≤ 2.53)=0.4943 By symmetry of the normal curve, this is also the area to the LEFT of the mean down to a value of z = –2.53 Then P (-2.53 ≤ z ≤ 2.53) = 0.4943 + 0.4943 = 0.9886 Continuous Random Variables By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Calculating P(z  -1):

Calculating P ( z  -1) An example of finding the area under the standard normal curve to the right of a negative z value Shown is finding the under the standard normal for z ≥ -1 Continuous Random Variables By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Calculating P(z  1):

Calculating P ( z  1) An example of finding tail areas Shown is finding the right-hand tail area for z ≥ 1.00 Equivalent to the left-hand tail area for z ≤ -1.00 Continuous Random Variables By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Finding Normal Probabilities:

Finding Normal Probabilities General procedure: 1. Formulate the problem in terms of x values. 2. Calculate the corresponding z values, and restate the problem in terms of these z values 3. Find the required areas under the standard normal curve by using the table Note: It is always useful to draw a picture showing the required areas before using the normal table Continuous Random Variables By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Finding Z Points on a Standard Normal Curve:

Finding Z Points on a Standard Normal Curve Continuous Random Variables By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Finding a Tolerance Interval:

Finding a Tolerance Interval Finding a tolerance interval [   k] that contains 99% of the measurements in a normal population Continuous Random Variables By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Normal Approximation to the Binomial:

Normal Approximation to the Binomial The figure below shows several binomial distributions Can see that as n gets larger and as p gets closer to 0.5, the graph of the binomial distribution tends to have the symmetrical, bell-shaped, form of the normal curve Continuous Random Variables By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Normal Approximation to the Binomial Continued:

Generalize observation from last slide for large p Suppose x is a binomial random variable, where n is the number of trials, each having a probability of success p Then the probability of failure is 1 – p If n and p are such that np  5 and n (1 – p )  5, then x is approximately normal with Normal Approximation to the Binomial Continued Continuous Random Variables By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

The Exponential Distribution #1:

The Exponential Distribution #1 Suppose that some event occurs as a Poisson process That is, the number of times an event occurs is a Poisson random variable Let x be the random variable of the interval between successive occurrences of the event The interval can be some unit of time or space Then x is described by the exponential distribution With parameter l , which is the mean number of events that can occur per given interval Continuous Random Variables By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

The Exponential Distribution #2:

The Exponential Distribution #2 If l is the mean number of events per given interval, then the equation of the exponential distribution is The probability that x is any value between the given values a and b ( a < b ) is and

The Exponential Distribution #3:

The Exponential Distribution #3 The mean m X and standard deviation s X of an exponential random variable x are Continuous Random Variables By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

The Cumulative Normal Table:

The Cumulative Normal Table The cumulative normal table gives the area under the standard normal curve below z Including negative z values The cumulative normal table gives the probability of being less than or equal any given z value See Table 5.3 Also see Table A.19 in Appendix A Continuous Random Variables By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

The Cumulative Normal Table:

Most useful for finding the probabilities of threshold values like P ( z ≤ a ) or P ( z ≥ b ) Find P ( z ≤ 1) Find directly from cumulative normal table that P ( z ≤ 1) = 0.8413 Find P ( z ≥ 1) Find directly from cumulative normal table that P ( z ≤ 1) = 0.8413 Because areas under the normal sum to 1 P ( z ≥ 1) = 1 – P ( z ≤ 1) so P ( z ≥ 1) = 1 – 0.8413 = 0.1587 The Cumulative Normal Table Continuous Random Variables By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

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M.Phil (Statistics)   GC University, . (Degree awarded by GC University)   M.Sc    (Statistics)   GC University, . (Degree awarded by GC University) Statitical Officer (BS-17 ) ( Economics & Marketing Division ) Livestock Production Research Institute Bahadurnagar ( Okara ), Livestock & Dairy Development Department, Govt. of Punjab Name                                        Shakeel Nouman Religion                                  Christian Domicile                            Punjab (Lahore) Contact #                            0332-4462527. 0321-9898767 E.Mail                                 [email protected] [email protected] Continuous Random Variables By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

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