Probability distribution • The probability distribution shows how the set of all possible mutually exclusive events is distributed. • It can be regarded as the theoretical equivalent of an empirical relative frequency distribution, with its own mean and variance. • It comprises all the values that the random variable can take, with their associated probabilities. Dr. R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
Binomial distribution or Bernoullian distribution It is a probability distribution expressing the probability of one set of alternatives, i.e. success or failure. Assumptions: 1. An experiment is performed under the same condition for a fixed number of trials, say ’n’ 2. In each such trial, there are only two possible chances of the experiment, success or failure 3. The probability of success denoted by ‘p’ which remains constant from trial to trial and the probability of failure denoted by q=1-p 4. The trials are independent Dr. R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
• In a series of n independent trials, if p is the constant probability of success at a single trial, then the variate ’x ’, i.e the number of success at these ‘n’ trials is said to follow binomial distribution. • The variate takes values from 0 to n (all integers), the probability of getting 0, 1, 2, …, n successes at these n trails is qn,nC1qn-1p, nC2qn-2p2,…, nCxqn-xpx, …, pn respectively, which are the respective terms of binomial expansion (q+p)n Dr. R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
Binomial distribution • Suppose if we have N sets of ‘n’ trials, the number of sets in which we will have 0, 1, 2, …, n success will be given by the successive terms of binomial expansion N(q+p)n. • Thus, we classify the sets according to the number of successes which they contain and we get a frequency distribution which is known as the binomial distribution Dr. R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
No. of success Frequency
0 1 Nqn nC1qn-1p
2 nC2qn-2p2
…
x nCxqn-xpx
...
n npn
Properties of Binomial distribution •It is a distribution of discontinuous or discrete variate •It has two parameters (constants).They are n and p, where n denotes the number of independent trials and p denotes the constant probability of success at a single trial • It takes values from 0 to n (all integers), i.e. 0, 1,2,…, n Dr. R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
Properties of Binomial distribution • Its mean is np and variance is npq, where q=1-p SD = √ npq; Variance is always less than mean • The different frequencies are different terms of binomial expansion N (q+p)n • It is symmetrical, when p=q=½ • When n is large and p is small such that np is constant, the binomial distribution tends to a Poisson distribution • When n is large and p=q=½, the binomial tends to become a normal distribution Dr. R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
Poisson distribution •
It is a discrete probability distribution and is limiting form of binomial distribution, when n is large and p is small such that np is constant. • It is a distribution of rare events. • It is also called as the law of improbable events Dr. R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
Formula- Poisson distribution P(x)
=
N e-m.mx x! where, m is the mean, N is the total frequency and x is the no. success
Dr. R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
Properties of Poisson distribution 1.It has one parameter m 2.It is a distribution of discrete variate 3.It takes values from 0 to Îą 4.Mean is approximately equal to variance 5.Skewness is 1/ m 6.Kurtosis is 3 + [1/ m]
Dr. R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
Normal Distribution or Normal Probability Distribution or Gaussian Distribution •
The normal distribution is the most useful theoretical distribution for continuous variable. • Discovered by De Movire as the limiting form of Binomial distribution and was also known to Laplace. • Gauss is the first one who made reference to this and it was erratically named after him. • The frequency curve corresponding to normal distribution is normal frequency curve or normal curve or Gaussian or Laplacian or probability curve. Dr. R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
2 N ( ) − x − m y= e 2σ2 σ 2π
where m = mean, N = total frequency, σ2 = variance Properties of Normal distribution 1. It is a distribution of continuous variates 2. The variate takes values from –α to+ α 3. It is symmetrical distribution; mean=median=mode 4.The slope of the curve is bell shaped. The ends of curve tails off asymptotically to the base 5. It has two parameters m and σ2 6.The first and the third quartiles are equi-distant from the median 7. MD = 4/5 σ or Dr.0.7979 σ R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
i.
mean ± 2/3 σ covers 50% of the observation
ii. mean ± σ covers 68.27 % of the observation iii. mean ± 2σ covers 95.45% of the observation •All odd moments about mean = 0 iv. mean ± 3σ covers 99.73% of the observation •Skewness is zero •Kurtosis is 3. It is mesokurtic. A frequency curve is leptokurtic if kurtosis > 3 A frequency curve is platykurtic if kurtosis < 3
Dr. R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
• If mean =0; SD=1 then the normal distribution is a standard normal distribution and its equation is
N y= e 2π
− x2
Dr. R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
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