PROBABILITY DISTRIBUTION Probability Distribution Gallery RANDOM VARIABLE OUT
PROBABILITY DISTRIBUTION
Probability Distribution Gallery
RANDOM VARIABLE • OUT COME – CAN BE PREDETERMINED BUT CAN NOT BE PRE-FORCED • TAKES ON DIFFERENT VALUES AS A RESULT OF THE OUTCOMES OF A RANDOM EXPERIMENT. • VALUE CHANGES FROM ONE OCCURANCE TO ANOTHER IN AN UNPREDICTABLE SEQUENCE.
EXPECTED VALUE • If the prob of obtaining the amounts, a 1, a 2, a 3……. . an are p 1, p 2, p 3…. . pn respectively, • Then expected value, ‘E’ = a 1. p 1+a 2. p 2+…………+an. pn E= or a i * pi i = 1 to n
ILLUSTRATIONS Q 4 If it rains, an umbrella salesman can earn Rs 30 per day. If weather is fair then he can lose Rs 6 per day. What would be his expectation if the prob of rain is 0. 3?
POSSIBLE OUTCOMES FROM TWO TOSSES OF A COIN FIRST TOSS Second TOSS No of TAILS Prob of Outcomes T T 2 . 5 X. 5=. 25 T H 1 . 5 X. 5=. 25 H T 1 . 5 X. 5=. 25 H H 0 5 X. 5=. 25 TOTAL 1
PROB DIST of POSSIBLE No of TAILS TOSSES Prob of Getting Tail 2 TT . 25 1 H T, TH (. 25 +. 25)=. 5 0 HH . 25 TOTAL 1
PROBABILTY DISTRIBUTION 0, 6 0, 5 0, 4 0, 3 0, 2 0, 1 0 0 1 2
PROB DISTRIBUTION Freq distribution? Listing all possible outcomes of an experiment, then indicating observed freq of each possible outcome Prob Distribution ? Listing the probabilities of all possible outcomes that could result if the experiment was done.
PROB DISTRIBUTION § Types of Probability Distribution. § Discrete Distribution – Can take on only limited values § Continuous Distribution – can take on any value in the given range
PROB DISTRIBUTION § Types of Probability Distribution. § Discrete Distribution § Binomial Distribution § Poisson Distribution § Continuous Distribution § Normal Distribution
BINOMIAL PROB DISTR
HOW TO ANSWER Qs? What is the prob of at least one hit in 30 rounds? What is the prob of exactly seven hits in 80 rounds?
TOSS OF A COIN. 5 T. 5 H H H 5 . 5 T . H H T. 5 TOSS 1 T 5 TOSS 2 H . 5 T . 5 . 125 (HHH). 125 (HHT). 125 (HTH). 125 (HTT). 125 (THH). 125 (THT). 125 (TTH). 125 (TTT) TOSS 3 P (0 H) =. 125 ; P (1 H) =. 375 ; P (2 H) =. 375 ; P (3 H) =. 125
CHARACTERISTICS OF BINOMIAL DISTRIBUTION v A discrete distribution. v No of trials is pre-determined. v Each trial must result in only two possible outcomes, ie either success (prob ‘p’) or failure (prob ‘q’ = 1 - p). v Prob of success in each trial must be same. v Trials must be statistically independent. v In such sit, if number of trials = n; prob of success = p; prob of failure = q = (1 -p), then, Prob of exactly ‘r’ successes = ncr pr q n-r
Binomial Distribution Formula
BINOMIAL DISTRIBUTION (BD) Ø Prob of exactly ‘r’ successes = ncr pr q Ø Mean of the BD is = ‘np’. Ø Standard Deviation is = (npq) n-r Ø The graph of BD is symmetrical (for p = 0. 5) and skewed (for p 0. 5). Øn. C r = n!/r!*(n-r)!
POISSON DISTRIBUTION
POISSON DISTRIBUTION? • Opportunity for Occurance High • Probability Low • Mean Determinable or available
SOME ILLUSTRATIONS No of earthquakes in an area. No of road accidents at a crossing. No of encounters with terrorists in a sector. No of crash landings at an airstrip. Collision of Ships while entering Harbour
CHARACTERISTICS OF POISSON PROB DISTN Ø Discrete type of distribution. Ø Potential for occurrence very high. Ø Prob of occurrence low (<=0. 1). Ø Occurrences statistically independent. Ø Mean No of occurrences in unit time or space proportional to the size of unit.
SHAPE OF POISSON DISTRIBUTION Y prob 0 k X
POISSON DISTRIBUTION If is mean number of occurrence per unit interval of time/space for a particular event Then prob that the event will occur exactly ‘k’ times is given by: e- * k P (k) = k ! where ‘e’ = Napier constant = 2. 7182
Shape of the Distribution
NORMAL DISTRIBUTION
POISSON DISTRIBUTION AS AN APPROXIMATION OF THE BINOMIAL DISTRIBUTION • The Poisson’s distribution can be a reasonable approximation of the Binomial under certain conditions like – n is indefinitely large – p for each trial is indefinitely small – np = m is finite
NORMAL DISTRIBUTION • Binomial and Poisson distribution for discrete random variable • Mathematical distribution for continuously varying variable is required • Normal distribution • Most useful theoretical distribution for continuous variable
BINOMIAL DISTRIBUTION USING EXCEL BINOMDIST how All Returns the individual term binomial distribution probability. S Use BINOMDIST in problems with a fixed number of tests or trials, when the outcomes of any trial are only success or failure, when trials are independent, and when the probability of success is constant throughout the experiment. Syntax BINOMDIST(number_s, trials, probability_s, cumulative) Number_s is the number of successes in trials. Trials is the number of independent trials. Probability_s is the probability of success on each trial. Cumulative is a logical value that determines the form of the function. If cumulative is TRUE, then BINOMDIST returns the cumulative distribution function, which is the probability that there at most number_s successes; if FALSE, it returns the probability mass function, which is the probability that there are number_s successes. Remarks • Number_s and trials are truncated to integers. • If number_s, trials, or probability_s is nonnumeric, BINOMDIST returns the #VALUE! error value. • If number_s < 0 or number_s > trials, BINOMDIST returns the #NUM! error value. • If probability_s < 0 or probability_s > 1, BINOMDIST returns the #NUM! error value.
Check Also !! CRITBINOM(trials, probability_s, alpha) • Trials is the number of Bernoulli trials. • Probability_s is the probability of a success on each trial. • Alpha is the criterion value. Returns the smallest value for which the cumulative binomial distribution is greater than or equal to a criterion value. Use this function for quality assurance applications. For example, use CRITBINOM to determine the greatest number of defective parts that are allowed to come off an assembly line run without rejecting the entire lot.
POISSON DISTRIBUTION USING EXCEL Returns the Poisson distribution. A common application of the Poisson distribution is predicting the number of events over a specific time, such as the number of cars arriving at a toll plaza in 1 minute. Syntax POISSON(x, mean, cumulative) X is the number of events. Mean is the expected numeric value. Cumulative is a logical value that determines the form of the probability distribution returned. If cumulative is TRUE, POISSON returns the cumulative Poisson probability that the number of random events occurring will be between zero and x inclusive; if FALSE, it returns the Poisson probability mass function that the number of events occurring will be exactly x.
NORMAL DISTRIBUTION • Described in 1733 by de Moivre as an approximation to the binomial distribution when the number of trails is large • Derived in 1809 by Gauss Abraham de Moivre (16671754) • Importance lies in the Central Limit Theorem, which states that the sum of a large number of independent random variables (binomial, Poisson, etc. ) will approximate a normal distribution – Example: Human height is determined by a large number of factors, both genetic and environmental, which are additive in their effects. Thus, it follows a normal distribution Karl F. Gauss (1777 -1855)
NORMAL (GAUSSIAN) DISTRIBUTION • It is an approximation to binomial distribution • Whether or not p is equal to q, n becomes large • Tails stretch infinitely in both directions
THE NORMAL DISTRIBUTION • A continuous random variable is said to be normally distributed with mean and variance 2 , if its probability density function is 1 (x )2/2 2 P(X) = e 2
NORMAL (GAUSSIAN) DISTRIBUTION • A single normal curve exists for any combination of , – these are the parameters of the distribution and define it completely • The normal distribution can have different shapes depending on different values of , , but there is one and only one normal distribution for any given pair of values for and
NORMAL DISTRIBUTION • Normal distribution limiting case of binomial distribution when N is too large and neither p nor q is very small • Normal distribution limiting case of Poisson distribution when mean is large • The mean of a normally distributed population lies at the centre of its normal curve • Two tails of the normal probability distribution extend indefinitely and never touch horizontal axis • Positive probability for continuous random variable
NORMAL DISTRIBUTION: PROPERTIES • For continuous variable • Curve is bell shaped and symmetrical appearance, mean and median coincide in • Normal distribution is defined by its , • Every normal distribution has its own , • The mean, median distribution are equal and mode of normal
NORMAL DISTRIBUTION: PROPERTIES • The area under the curve is 1 • Distribution denser in the center and less so in the trails • Trails never touch the axis, range is unlimited in both directions • Normal curve is unimodal, has only one mode
NORMAL DISTRIBUTION: PROPERTIES • The area under the normal curve is distributed as – Mean + 1 covers 68. 27% area, 34. 135 area on either side of the mean – Mean + 2 covers 95. 45% area – Mean + 3 covers 99. 73% area
68 -95 -99. 7 Rule 68% of the data 95% of the data 99. 7% of the data
CONDITIONS FOR NORMALITY • The casual forces must be numerous and of approximately equal weight • The forces must be same over the universe from which observations are drawn (condition of homogeneity) • The forces affecting events must be independent of one another • The operation of the casual forces must be such that deviation above the population mean are balanced as to magnitude and number by deviations below the mean (conditions of symmetry)
AREA RELATIONSHIP § The area of the normal curve between § Mean Ordinate § Ordinate at various sigma distance from the mean (as percentage of the total area) 42
STANDARD NORMAL DISTRIBUTION
AREA UNDER NORMAL CURVE § How to convert Normal curve with mean x and standard deviation By performing § Change of scale from X to Z Scale § Change of origin σ § In original scale X – scale mean and standard deviation are µ & § In new scale Z-scale mean = 0 and SD = 1
AREA UNDER NORMAL CURVE §The formula that enables us to change from X-scale to Zscale and vice versa § Z = (X – μ) / σ § In practice no matter what units of measurement the normal random variable x has ( kg , rupees, cm , hours etc) we will always be able to convert it into a standard scale by transformation formula & then determining the desired probabilities from the table of standard normal distribution
AREA UNDER NORMAL CURVE § The transformation from X to Z is termed as Ztransformation. § Given a value of X , the corresponding Z value tell us : § How far away § What direction X is from its mean in terms of its standard deviation σ Z = 1. 8 means that the value σ to the right of mean x of X is 1. 8
AREA UNDER NORMAL CURVE • Area under the curve , corresponding to a normal distribution equal to unity, regardless of particular number of observations involved • We thus have a Normal Distribution that is independent of • Number of observations • Mean • Standard deviation • This is known as UNIT NORMAL DISTRIBUTION OR STANDARD NORMAL DISTRIBUTION • It is applicable to any distribution that is normal regardless of mean, SD & No of Observations
RELATION BETWEEN THREE DISTRIBUTIONS § Binomial (large ‘n’ and small ‘p’) → Poisson Distribution § Poisson (large ‘m’) → Normal Distribution 48
NORMAL DISTRIBUTION EXAMPLE § Given is the mean height of soldiers as 68. 22 inches, with variance of 10. 8 inches. How many soldiers are expected to be above 72 inches out of strength of 1000? § Assuming distribution is normal, following values are given: X = 72; μ = 68. 22; σ2 = 10. 8
NORMAL DISTRIBUTION EXAMPLE § σ2 = 10. 8, so σ = 3. 286 § Hence, Z = (X – μ) / σ = 1. 16 § Note the value from the table for ‘Z’, which is 0. 3749 § Probability of 72 inches or more height is required, so area to its right is to be found § Area to the right of this value is: 0. 5000 – 0. 3749 = 0. 1251 which is the required probability
NORMAL DISTRIBUTION EXAMPLE – 1 § Hence, no. of soldiers above 72 inch height out of 1000 strength will be: 0. 1251 x 1000 = 125. 1 ≈ 125 68. 22 0. 3749 0. 1251 72. 0
NORMDIST USING EXCEL Returns the normal distribution for the specified mean and standard deviation. This function has a very wide range of applications in statistics, including hypothesis testing. Syntax NORMDIST(x, mean, standard_dev, cumulative) X is the value for which you want the distribution. Mean is the arithmetic mean of the distribution. Standard_dev is the standard deviation of the distribution. Cumulative is a logical value that determines the form of the function. If cumulative is TRUE, NORMDIST returns the cumulative distribution function; if FALSE, it returns the probability mass function. 52
NORMINV Returns the inverse of the normal cumulative distribution for the specified mean and standard deviation. Syntax NORMINV(probability, mean, standard_dev) Probability is a probability corresponding to the normal distribution. Mean is the arithmetic mean of the distribution. Standard_dev is the standard deviation of the distribution. 53 BPKC : CLM (STATS)
NORMSDIST Returns the standard normal cumulative distribution function. The distribution has a mean of 0 (zero) and a standard deviation of one. Use this function in place of a table of standard normal curve areas. Syntax NORMSDIST(z) Z is the value for which you want the distribution. BPKC : CLM (STATS) 54
NORMSINV Returns the inverse of the standard normal cumulative distribution. The distribution has a mean of zero and a standard deviation of one. Syntax NORMSINV(probability) Probability is a probability corresponding to the normal distribution. Remarks • If probability is nonnumeric, NORMSINV returns the #VALUE! error value. • If probability < 0 or if probability > 1, NORMSINV returns the #NUM! error value. Given a value for probability, NORMSINV seeks that value z such that NORMSDIST(z) = probability. Thus, precision of NORMSINV depends on precision of NORMSDIST. NORMSINV uses an iterative search technique. If BPKC : CLM (STATS) the search has not converged after 100 iterations, the function returns the 55 #N/A error value.
CHEBYSHEV’S THEOREM • For any set of data at least of the data elements must lie within ‘k’ standard deviations on either side of the mean. • Where k is any number > 1. Therefore, standard deviation enables us to determine, with a great deal of accuracy, where the values of a freq distribution are located, in relation to the mean.
? BPKC : CLM (STATS) 57
CHEBYSHEV’S THEOREM • For any set of data at least of the data elements must lie within ‘k’ standard deviations on either side of the mean. • Where k is any number > 1. Therefore, standard deviation enables us to determine, with a great deal of accuracy, where the values of a freq distribution are located, in relation to the mean.
NORMAL DISTRIBUTION EXAMPLE – 1 § Hence, no. of soldiers above 72 inch height out of 1000 strength will be: 0. 1251 x 1000 = 125. 1 ≈ 125 68. 22 0. 3749 0. 1251 72. 0
NORMDIST USING EXCEL Returns the normal distribution for the specified mean and standard deviation. This function has a very wide range of applications in statistics, including hypothesis testing. Syntax NORMDIST(x, mean, standard_dev, cumulative) X is the value for which you want the distribution. Mean is the arithmetic mean of the distribution. Standard_dev is the standard deviation of the distribution. Cumulative is a logical value that determines the form of the function. If cumulative is TRUE, NORMDIST returns the cumulative distribution function; if FALSE, it returns the probability mass function. 66
NORMINV Returns the inverse of the normal cumulative distribution for the specified mean and standard deviation. Syntax NORMINV(probability, mean, standard_dev) Probability is a probability corresponding to the normal distribution. Mean is the arithmetic mean of the distribution. Standard_dev is the standard deviation of the distribution. BPKC : CLM (STATS) 67
NORMSDIST Returns the standard normal cumulative distribution function. The distribution has a mean of 0 (zero) and a standard deviation of one. Use this function in place of a table of standard normal curve areas. Syntax NORMSDIST(z) Z is the value for which you want the distribution. BPKC : CLM (STATS) 68
NORMSINV Returns the inverse of the standard normal cumulative distribution. The distribution has a mean of zero and a standard deviation of one. Syntax NORMSINV(probability) Probability is a probability corresponding to the normal distribution. Remarks • If probability is nonnumeric, NORMSINV returns the #VALUE! error value. • If probability < 0 or if probability > 1, NORMSINV returns the #NUM! error value. Given a value for probability, NORMSINV seeks that value z such that NORMSDIST(z) = probability. Thus, precision of NORMSINV depends on precision of NORMSDIST. NORMSINV uses an iterative search technique. If the search has not converged after 100 iterations, BPKC : CLM (STATS) the function 69 returns the #N/A error value.
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