Binomial distribution* One widely used probability distribution of a discrete random variable.*Named after Swiss mathematician Jacob Bernoulli.*Known as the outcome of Bernoulli process.* Binomial distribution describes discrete, not continuous ,data , resulting from an experiment known as Bernoulli process.

Binomoial distribution the process includes_1.The process is performed under the same conditions for a fixed & finite number of trials,n.2.Each trial is independent from other trials.3.Each trial has only two possible outcome4.The probablity of success,p,remains constant from trial to trial.

ProblemIf we toss a coin 3 times then chances of getting exactly 2 heads will be_As, p=0.5q=.05r= 2n=3

p+q=1

prolem

Binomial distribution

Condition for Binomial distribution

n=number of trialp=probability of success

Mean of binomial

Mean- binomial

n=number of trialsp=probability of successq=probability of failure

Variance-binomial

Variance- binomial

Problem1. In a certain factory10% of the products are defective. If a random sample of 25 products are drawn from the output. Find the probability that:a) Five or fewer will defectiveP(X 5) = p(x=0)+ p(x=1) + p(x=2) + p(x=3) + p(x=4) + p(x= 5) =.9666b) Two, three or four will be defectiveP(2 X 4)=P(X 4) P(X 1) = p(x=0)+ p(x=1) + p(x=2) + p(x=3) +p(x=4) (p(x=0)+ p(x=1) ) =0.9020-0.2712 =0.6308c)At least 2 will be defectived) At most 3 will be defective

Problem2. About 30% of the production are defective in a factory. 10 bulbs are packed in one packet. Among the 1000 packets find the probability that i) no bulb is defective ii) at best one bulb is defective and find such kind of expected number of packets.

* We can make the following generalizations:# When p is small (0.1), the distribution is skewed to the right.# As p increases (0.3), the skewness is less noticeable.# When p= 0.5 the distribution is symmetrical# When p is larger than 0.5 the distribution is skewed to the left.# This is true for any pair of complemetary p and q values (0.3 and .7, 0.4 and 0.6, and 0.2 and 0.8).

POISSON DISTRIBUTION The second important discrete probability distribution named after the French mathematician Simeon Denis Poisson(1781-1840).Poisson distribution differs from the binomial dist in two aspects:Rather than consisting of discrete trials, the dist operates continuously over some given amount of time, distance, area etc.Rather than producing a sequence of uccess and failures, the dist produces successes, which occur at random points in the specified time, distance, area.

Poisson distribution

Characteristics of Poisson Distribution1)The occurrence of the events is independent. That is the occurance of an event in an interval of space or time has no effect on the probability2)An infinite number of occurrences of the event must be possible in the interval3)The probability of single occurrence in a given interval is proportional to the length of the interval4)In any infinitesimal(extremely small) portion of interval,the probability of two or more occurrences of the event is negligible.

Differs from the binomial distribution in important aspects1)Rather than consisting of discrete trials,the distribution operates continuously over some given amount of time, distance, area etc2)Rather than producing a sequence of successes & failures.The distribution produces successes which occur at random points in the specified time,distance,area.3)The mean &variance of poisson distribution are Same.

Conditions for which Poisson Distribution as an approximation of the Binomial Distribution:1)When the number of trials are large that means n is large2)The Binomial Probability of success is small that means p is small

* The rule most often used by statisticians is that the Poisson is a good approximation of Binomial when n is greater than or equal to 20 & p is less than or equal to 0.05

Business application of Poisson Distribution:

1)The demand for a product2)Typographical errors occurring on the pages of a book3)The occurrence of accident in a factory4)The arrival pattern in a departmental store5)The occurrences of flaws in a bolt in a factory6)The arrival of calls at a switch board

problem1. If 3% of the electric bulbs manufactured by a company are defective, find the probability that in a sample of 100 bulbs i) 1 bulbs ii) at most 2 bulbs will be defective.

2. What probability model is appropriate to describe a situation where 100 misprints are distributed randomly throughout the 100 pages of a book? For this model, what is the probability that a page observed at random will contain at least three misprints?M=np=1. e-m = .3679P[x>/ 3]= .0802

Normal distribution Normal distribution is the most important continuous probability distribution. It is continuous means it accept all the value between a given range. The probability of single point is not possible. It is between fxdx=1 -

Properties of normal distributionThe curve has a single peak ,one max point thus it is unimodal.It is symmetrically distributed about the mean,( skewness=0) If the curve is folded along its vertical axis, the two halves will concideThe no of cases below the mean and above the mean is same, which makes mean and median same. Also the height of the normal curve is at its maximum at the mean. Thus Mean, median, mode are all equal.Two tails normal distribution are extended forever and never touches horizontal axis.First and third quartiles are equidistantAll odd moments are zero, skewness=0, kurtosis=3

Properties of normal distribution*Areas under the normal curve: i)Approximately 68% of all the values in a normally distributed population led within 1 standard deviation from the mean. ii)Approximately 95.5% of all the values in a normally distributed population led within 2 standard deviation from the mean. iii) Approximately 99.7% of all the values in a normally distributed population led within 3 standard deviation from the mean.

Distribution function of Normal :

Standard normal distribution:

Problems1. The average daily sales of 500 branch offices was tk 150 thousad and the sd tk 15thousand. Assuming the dist to be normal, indicate how many branches have sales between Tk 120 thous and tk 145 thousand2. The weight of a certain type of a car tyre is normally distributed with a mean of 45 kg and a variance of 4 kg. A random sample of 100 tyres is selected. What is the probability that the mean of this sample lies between 42.5 and 46.4 kg?3. A workshop produces 2000 units per day. The average weight of units is 130 kg with a sd of 10 kg. assuming normal distribution, how many units are expected to weight less than 142 kg?4. In a normal distribution, 31% of the items are under 45 and 8% are over 64. Find the mean and sd of the distributions.

problem

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