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A Probability Distribution is a rule that assigns probabilities to each element of a set of events that may occur. Types of Probability Distribution
A Probability Distribution is a rule that assigns probabilities to each element of a set of events that may occur. Types of Probability Distribution
If a random variable is a discrete variable, its probability distribution is called a discrete probability distribution . Ex , Suppose if we toss a coin are two times. This simple experiment can have four possible outcomes: HH, HT, TH, and TT. Now, Now, let the random variable X represent the number of Heads that result from this experiment. The random variable X can only take on the values 0, 1, or 2, so it is a discrete random variable. 0
1/4
1
2/4
2
1/4
Continuous Probability Distribution Continuous Probability Distribution is that in which the random variable can be any number within some given range of values.
5-5.5
4/10
5.5-5.8
3/10
5.8-6.00
3/10
A Discrete Probability Distribution is known as Probability mass function( p.m.f) and a continuous Probability Distribution is known as Probability density function( p.d.f),
1.Binomial 2. Poisson
3. Normal
It is the theoretical distribution of discrete random variables. It was discovered by mathematician James bernouli.
It is the discrete Probability Distribution.
The Binomial Distribution is associated with the name of J.Bernoulli. The features of Binomial Distribution is as follows: It is a random process which is performed under the same conditions for a fixed i) and finite number of trials, say n. ii)
Each trial is independent of other trials, i.e. probability of an outcome for any particular trial is not influenced by the outcomes of the other trials.
iii)
Each trial has two possible outcomes, which are mutually exclusive, such as ‘success’ or ‘failure’, ‘good’ or ‘defective’, ‘hit’ or ‘miss’, survive or die.
iv)
The probability of success , p , remains constant from trial to trial . q = The probability of failure, where q = 1-p.
These above conditions will satisfy if we toss a coin.
A fair coin is tossed 3 times and we are interested in finding the probability of exactly two heads. Therefore we will consider head as success and tail as failure with corresponding probabilities p and q respectively.
HHH, TTT, HTT , TTH, THT, HHT,HTH THH
HHT,HTH THH
P = 3/8.
As the number of tosses increases(say 20 0r 50 times), it becomes more and more difficult to calculate the probability. Here an easy method is required and hence we use binomial formula.
f( x) =P(x)= n C x p x q n-x n = the number of trials p= the probability of a success on a trial q = the probability of a failure on a trial X = the number of successes in n trials x = 0, 1, 2, . . ., n
If 3 balls are drawn at random one by one with replacement from a bag containing 8 white and 15 red balls. balls are drawn without replacement
1.
A coin is tossed six times , what is the probability of obtaining four or more heads?
2.
The incidence of a certain disease is such that on the average 20% of workers suffer from it. (Assuming the distribution fits binomial) If 10 workers are selected at random, find the probability that i) exactly 2 workers suffer from the disease ii) not more than 2 workers suffer from the disease.
Q3. It is believed that 20% of the employees in an office are usually late. If 10 employees report for duty on a given day, what is the probability that: (a) Exactly 3 employees are late. (b) At most 3 employees are late. (c) At least 3 employees are late.
0.2, 0.88, 0.32
.
Q4.A company manufactures motor parts. The market practice is such that goods are sold on one month’s credit in the domestic market. The limit of credit sales of different buyers is decided by the manufacturer based upon the perception of the goodwill of the individual buyer. The manufacturer observed that 30% of the buyers take more than a month in making the payment. In a particular city if goods are sold to 5 buyers on credit, what is the probability that (i)
Exactly 3 buyers will delay the payment beyond one month
(ii)
At most 2 buyers will delay ?
0.1323, 0.837
Q5. After the privatisation of the power sector in Delhi, consumers often complain that new meters installed by the private power companies are defective and run faster. On testing of meters it was found that 10% of the meters were defective and run faster. Ina housing society, a test check was conducted on 6 meters, what is the probability that (i) one meter is defective; (ii) at least one meter is defective?
0.354, 0.47
Let n denotes no. of trials in a binomial distribution . p denotes probability of success and q denotes probability of failure. x takes the values 0,1,2,3….n.
0
nc q n.p0 0
0
1
nc qn-1.p 1
1. n.qn-1.p
2
nc qn-2.p2 2
2.n(n-1) /2qn-2.p2
. . . n
nc pn n
q0
npn
Arithmetic Mean= ∑x. P(x) ∑P(x)
=np
np( p+q) =np(1) n-1 = np
n-1
Mean= np Variance=npq
Q1.In eight throws of a die 1 or 6 is considered as success. Find the mean number of success and the SD. Q2.The mean of a binomial distribution is 40 and standard deviation is 6. Calculate n, p and q. Q3.Bring out the fallacy , if any, in the following statement. The mean is 10 and its s.d. is 6.
Q4. The probability of a bomb hitting a target is 2/5. Two bombs are enough to destroy a bridge. If 7 bombs are dropped at the bridge , find the probability that the bridge is destroyed.
Q5. A student appearing in a multiple choice test answers 10 questions, purely by guessing. If there are 5 choices for each question, What is the probability that 6 or more answers will be correct. 0.007.
A second important discrete probability Distribution is the Poisson Distribution, named after the French mathematician S. Poisson. This distribution is used to describe the behaviour of events, where the total no of observation (n)is large and their chance of Occurrence(p) is low.
The number of accidents that occur on a given highway during a given time period. The number of printing mistakes in a page of a book The number of earthquakes in Delhi in a decade. The number of deaths of the policyholders recorded by the LIC . It is used by the quality control departments of manufacturing industries to count the number of defects found in a lot.
Poisson distribution is given by,
P(x) = e-m.mx x! e=base of natural logarithm whose value is 2.7183 x= no. of occurrences of an event, x =0,1,2,3, m= mean
= np and m >0
Mean= m Variance=m
On the average, one in 400 items is defective. If 100 items are packed in each box, what is the probability that any given box will contain : no defective i) less than two defectives ii) iii) one or more defectives iv) more than 3 defectives
Here, p = 1/400, probability of defective item which is very low. n= 100 ,no of items packed in the box which is quite large. m= np = 0.25 ; average number of defectives in a box of 100 items.
i) P (x=0) = e-m.mx = x! ii) P (x<2) = P (x=0) + P (x=1)
Customers arrive randomly at a retail counter at an average rate of 10 per hour. Assuming a Poisson Distribution, calculate No customer arrives in any particular minute. i) Exactly one customer arrives in any particular minute. ii)
Hint: m= 10
Assuming that the probability of a fatal accident in a factory during a year is 1/1200, calculate the probability that in a factory employing 300 workers, there will be at least two fatal accidents in a year.
Hint: e
-0.25
=0.7787
A manufacturer , who produces medicine bottles, finds that 0.1% of the bottles are defective. The bottles are packed in boxes containing 500 bottles. A drug manufacturer buys 100 boxes from the producer of bottles. Using Poisson distribution find out how many boxes will contain (i) No defective (ii) At least 2 defectives ( e - 0.5 = 0.6065) Ans61 &9
If the probability of a defective bolt is 0.2, find (i) The mean (ii) The standard deviation in a total of 900 bolts.
13.4
(i)
n
(ii)
p np=m
(iii)
α
0
X 0 1 2 3 4
p=0.4, q= 0.6
P(X) 1 . (.4)0 . (.6)4 = .130 4 . (.4)1 . (.6)3 = .346 6 . (.4)2 . (.6)2 = .346 4 . (.4)3 . (.6)1 = .154 1 . (.4)4 . (.6)0 = .026
0.35
X 0 1 2 3 4
0.3 0.25 0.2 0.15 0.1 0.05 0 0
1
2
3
4
n=4 , p=.05 P(X) .130 .346 .346 .154 .026
When no of trials become infinite, X takes continuous values, the curve becomes smooth, called normal distribution.
Hence, normal distribution is approximation to binomial distribution. μ = Mean of the normal distribution. σ= standard deviation of the normal distribution. A random variable follows normal distribution with mean μ and standard deviation σ. X ~ N (μ , σ )
- ∞