Successfully reported this slideshow.
Upcoming SlideShare
×

of

27

Share

# Bernoullis Random Variables And Binomial Distribution

Bernoullis Random Variables And Binomial Distribution

See all

See all

### Bernoullis Random Variables And Binomial Distribution

1. 1. 1.10 Bernoulli’s random Variables & Binomial Distribution<br />
2. 2. Bernoulli Random Variable<br />Suppose that a trial, or an experiment, whose outcome can be classified as either a success or a failure is performed. If we let X=1 when the outcome is a success and X=0 when the outcome is a failure, then the pmf of X is given by<br />
3. 3. Bernoulli Random Variable<br /> A random variable X is said to be a Bernoulli random variable (after the Swiss mathematician James Bernoulli) if its probability mass function is given by <br />
4. 4. Binomial Random Variable<br />Suppose now that n independent trials, each of which results in a success with probability p and in a failure with probability 1-p, are to be performed. If X represents the number of successes that occur in the n trials, then X is said to be a Binomial random variable with parameters (n,p) . Thus a Bernoulli random variable is just a binomial random variable with parameters (1,p) .<br />
5. 5. Binomial Distribution<br />Bernoulli Trials<br />There are only two possible outcomes for each trial.<br />The probability of a success is the same for each trial.<br />There are n trials, where n is a constant.<br />The n trials are independent.<br />
6. 6. Binomial Distribution <br />Let X be the random variable that equals the number of successes in n trials.<br />If p and 1 – p are the probabilities of success and failure on any one trial then the probability of getting x successes and n – x failures in some specific order is <br />px(1- p)n – x<br />The number of ways in which one can select the x trials on which there is to be a success is <br />
7. 7. Binomial Distribution <br />Thus the probability of getting x successes in n trials is given by<br />This probability distribution is called the binomial <br />distribution because for x = 0, 1, 2, …, and n the <br />value of the probabilities are successive terms of <br />binomial expansion of [p + (1 – p)]n;<br />
8. 8. Binomial Distribution <br />for the same reason, the combinatorial quantities <br />are referred to as binomial coefficients. <br />The preceding equation defines a family of probability distributions with each member characterized by a given value of the parameterp and the number of trials n.<br />
9. 9. Binomial Distribution <br />Distribution function for binomial distribution<br />
10. 10. Binomial Distribution <br />The value of b(x;n,p) can be obtained by formula<br />since the two cumulative probabilities B(x; n, p) and B(x - 1; n, p) differ by the single term b(x; n,p).<br />If n is large the calculation of binomial probability can become quite tedious.<br />
11. 11. Binomial Distribution Function<br />Table for n = 2 and 3 and p = .05 to .25<br />
12. 12. Example<br />
13. 13. The Mean and the Variance of a Probability Distribution<br />Mean of discrete probability distribution<br />The mean of a probability distribution is the mathematical expectation of a corresponding random variable. <br />If a random variable X takes on the values x1, x2, …, or xk, with the probability f(x1), f(x2),…, and f(xk), its mathematical expectation or expected value is <br /> = x1· f(x1) + x2· f(x2) + … + xk· f(xk) <br />
14. 14. The Mean and the Variance of a Probability Distribution <br />Mean of binomial distribution<br />p  probability of success<br />n  number of trials<br />Variance of binomial distribution<br />
15. 15. The Mean and the Variance of a Probability Distribution <br />Mean of binomial distribution<br />p  probability of success<br />n  number of trials<br />Proof:<br />
16. 16. The Mean and the Variance of a Probability Distribution <br />Put x – 1= y and n – 1 = m, so n – x = m – y, <br />
17. 17. Computing formula for the variance<br />Variance of binomial distribution<br />Proof:<br />
18. 18. Put x – 1 = y and n – 1 = m <br />The Mean and the Variance of a Probability Distribution <br />
19. 19. The Mean and the Variance of a Probability Distribution <br />
20. 20. Put y – 1 = z and m – 1 = l in first summation<br />The Mean and the Variance of a Probability Distribution <br />
21. 21. Moment Generating function for Binomial distribution<br />
22. 22. Second ordinary/raw moment (moment about origin)<br />Moment Generating function for Binomial distribution<br />
23. 23. Moment Generating function for Binomial distribution<br />Moment Generating function for Binomial distribution<br />
• #### LingalaBhumikagoud

Jul. 30, 2021
• #### kusumakhushi

Nov. 11, 2020
• #### AakashSingh186

Apr. 17, 2020
• #### KhushbooGautam6

Apr. 16, 2020

Dec. 15, 2019

Nov. 3, 2019

Jan. 7, 2019

Jan. 4, 2019
• #### MariyaJeeja

Sep. 15, 2018
• #### lualkuch

May. 31, 2018

Mar. 8, 2018

Feb. 6, 2018
• #### ARRahman17

Jan. 15, 2018
• #### AlemayehuFikire

Dec. 25, 2017

Nov. 1, 2017

Aug. 3, 2017

Mar. 9, 2017

Dec. 9, 2016

May. 1, 2016
• #### SUJESHKR

Feb. 16, 2016

Bernoullis Random Variables And Binomial Distribution

Total views

11,505

On Slideshare

0

From embeds

0

Number of embeds

23

0

Shares

0