Probability Distribution
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Project by Ms.Mamta Barik Assistant Professor of JIMS management Technical Campus,Greater Noida. A probability distribution is a function that portrays the probability of acquiring the potential qualities that an irregular variable can assume. In other words the estimations of the variable fluctuate dependent on the hidden probability distribution.
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In probability theory and statistics, the binomial distribution is the discrete probability distribution gives only two possible results in an experiment, either Success or Failure. For example, if we toss a coin, there could be only two possible outcomes: heads or tails, and if any test is taken, then there could be only two results: pass or fail. This distribution is also called a binomial probability distribution.
Number of trials (n) is a fixed number.
The outcome of a given trial is either success or failure.
The probability of success (p) remains constant from trial to trial which means an experiment is conducted under homogeneous conditions.
The trials are independent which means the outcome of previous trial does not affect the outcome of the next trial.
Binomial Probability Distribution
In binomial probability distribution, the number of ‘Success’ in a sequence of n experiments, where each time a question is asked for yes-no, then the valued outcome is represented either with success/yes/true/one (probability p) or failure/no/false/zero (probability q = 1 − p). A single success/failure test is also called a Bernoulli trial or Bernoulli experiment, and a series of outcomes is called a Bernoulli process. For n = 1, i.e. a single experiment, the binomial distribution is a Bernoulli distribution.
There are two parameters n and p used here in a binomial distribution. The variable ‘n’ states the number of times the experiment runs and the variable ‘p’ tells the probability of any one outcome. Suppose a die is thrown randomly 10 times, then the probability of getting 2 for anyone throw is ⅙. When you throw the dice 10 times, you have a binomial distribution of n = 10 and p = ⅙.
The binomial distribution formula is for any random variable X, given by;
P(x:n,p) = nCx px (1-p)n-x
Where,
n = the number of experiments
x = 0, 1, 2, 3, 4, …
p = Probability of Success in a single experiment
q = Probability of Failure in a single experiment = 1 – p
The binomial distribution formula can also be written in the form of n-Bernoulli trials, where nCx = n!/x!(n-x)!. Hence,
P(x:n,p) = n!/[x!(n-x)!].px.(q)n-x
Binomial Distribution Mean and Variance
For a binomial distribution, the mean, variance and standard deviation for the given number of success are represented using the formulas
Mean, μ = np
Variance, σ2 = npq
Standard Deviation σ= √(npq)
Where p is the probability of success
q is the probability of failure, where q = 1-p
Properties of binomial distribution
The properties of the binomial distribution are:
• There are two possible outcomes: true or false, success or failure, yes or no.
• There is ‘n’ number of independent trials or a fixed number of n times repeated trials.
• The probability of success or failure remains the same for each trial.
• Only the number of success is calculated out of n independent trials.
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Probability Distribution
- 1. Presentation on PROBABILITY DISTRIBUTION Mamta Barik Applied Science Deptt. *Applied Mathematics-IV
- 2. * A probability distribution is a function that describes the likelihood of obtaining the possible values that a random variable can assume. In other words, the values of the variable vary based on the underlying is probability distribution. Binomial Poisson Normal They are of following types:
- 3. 1.BINOMIAL DISTRIBUTION This distribution is based on Bernoulli’s trials. It is a discrete distribution. It’s PDF is : P(x)= nCx px qn-x , 0<x<n where , n is total number of outcomes, x is possible number of outcomes, p is probability of success, q is probability of failure. Here, p+q=1 Mean=np Variance=npq
- 4. * Consider the following question:-
- 5. 2. POISSON DISTRIBUTION It is a discrete distribution based on Bernoulli’s trials. It is a limiting case of binomial distribution. When n becomes very large and P becomes very small then Binomial distribution tends to Poisson distribution. It’s PDF is: P(x) = e-λ λ^x / x! ,0<x<∞ * Poisson distribution has the following properties:- * Mean of the distribution = λ . * Variance of the distribution = λ .
- 6. * Consider the following question:-
- 7. 3. NORMAL DISTRIBUTION It is a continuous probability distribution whose Probability Mass Function(PMF) is: P(x)= [1/ -∞<x<∞ It’s Mean= μ Variance= σ2 Standard Deviation= σ It is a limiting case of Binomial distribution. When n becomes very large and P becomes close to ½ then B.D tends to Normal distribution. σ√2Π]*e[(-1/2)(x-μ/σ)2]
- 8. 3. NORMAL DISTRIBUTION It is a symmetrical distribution. In this, we convert x into z by the transformation: z= (x-μ)/σ Total area under the normal curve is unity. P(-∞<z<∞)= 1 P(-∞<z<0)=P(0<z<∞)= 0.5 -∞ z=0 ∞
- 9. * Consider the following question:-
- 10. THANK YOU!