8 binomial

Distribution Department of AGB Veterinary College, Hebbal, Bangalore

Distribution In the population, the values of the variables may be distributed according to some definite law of probability distribution known as theoretical distribution.

These distributions are based on the expectations on the previous experience.

Hence theoretical or probability distributions are mathematically deduced.

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Distributions The three important distributions are • Binomial distribution • Poisson distribution • Normal distribution

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Binomial distribution or Bernoulli’s distribution Definition: It is one of the discrete probability distribution.

It is used when there are exactly two mutually exclusive outcomes of a trial.

These outcomes are appropriately labeled Success and Failure.

The Binomial Distribution is used to obtain the probability of observing ‘r’ successes in ‘n’ trials, with the probability of success on a single trial denoted by ‘p’.

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

The requirements to be a binomial experiments are as follows: • •

• •

There must be a fixed number of trials.

Trials must be independent. One trial's outcome cannot affect the probabilities of other trials.

All outcomes of trials must be in one of two categories.

Probabilities must remain constant for each trial.

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Notations used S (success) and F (failure) The probabilities are represented as P(S) and P(F), respectively.

• P(S) = p (Success) • P(F) = q = 1-p (Failure) • n indicates the fixed number of trials.

• r indicates the number of successes (any whole number [0,n]).

• P (r) indicate the probability of getting exactly r successes in n trials.

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Formula P(X = r) = nCr p r (1-p)n-r n = Number of events. r = Number of successful events. p = Probability of success on a single trial. nCr = ( n! / (n-r)! ) r!

1-p = Probability of failure. or

P (r) = n! Prq n-r (n-r)!r!

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Example Toss a coin for 12 times. What is the probability of getting exactly 7 heads?

Step 1: Here, Number of trials n = 12 Number of success r = 7 (since we define getting a head as success) Probability of success on any single trial p = 0.5

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

P(X = r) = nCr p r (1-p)n-r n = 12

Step 2: • To Calculate nCr formula is used

nCr = ( n! / (n-r)! ) r! = ( 12! / (12-7)! ) 7! = ( 12! / 5! ) 7!

= 792 12x11x10x9x8x7x6x5x4x3x2x1 (5x4x3x2x1)(7x6x5x4x3x2x1) = 12x11x10x9x8 =792 5x4x3x2x1 Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

r=7

Step 3: Find pr. pr = (0.5)7 = 0.0078125 Step 4: To Find (1-p)n-r Calculate 1-p and n-r.

1-p = 1-0.5 = 0.5 n-r = 12-7 = 5 Step 5: Find (1-p) n-r. = (0.5)5 = 0.03125

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

P(X = r) = nCr p r (1-p)n-r n = 12

r=7

Toss a coin for 12 times. What is the probability of getting exactly 7 heads?

P(X = r) = nCr p r (1-p)n-r n = 12

Step 6: Solve P(X = r) = nCr p r (1-p) n-r = 792 × 0.0078125 × 0.03125 = 0.193359375

The probability of getting exactly 7 heads by tossing 12 times is 0.19

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

r=7

Example • A coin is tossed three times. Find the probability of getting exactly two heads.

Solution: This problem can be solved by looking that the sample space. There are three ways to get two heads.

HHH, HHT, HTH, THH, TTH, THT, HTT, TTT The answer is 3/8 or 0.375.

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Four requirements of a binomial experiment 1. There are only two outcomes for each trial - heads or tails.

2. There is a fixed number of trials (three).

3. The outcomes are independent of each other (the outcome of one toss in no way affects the outcome of another toss).

4. The probability of a success (heads) is 1/2 in each case.

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

In this case, n = 3, X = 2, p = 1/2,and q=1/2.

Hence, substituting in the formula gives

• which is the same answer obtained by using the sample space.

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Example • If a student randomly guesses at five multiple-choice questions, find the probability that the student gets exactly three correct. Each question has five possible choices.

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Solution • In this case n = 5, X = 3, and p = 1/5, since there is one chance in five of guessing a correct answer

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Poisson distribution • It is a distribution rare events • Also called as law of improbable events • Also known as limiting form of binomial distribution when ‘n’ is large and ‘p’ is small such that np is constant Example: No. of vehicles pass in in 100 minute during peak hour and lean hour

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Formula for Poisson distribution

m= mean N = total frequency X= no. of success

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Properties of Poisson distribution • • • • • •

It has one parameter m It is a distribution of discrete variate It takes values from 0 to ∞ Mean is approximately equal to variance Skewness is 1/m Kurtosis is 3 + (1/m)

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Normal distribution • Also known as normal probability distribution or Gaussian distribution • This is most useful theoretical distribution for continuous variable • The frequency curve is known as normal curve ot Gaussian or Laplacian or probability curve

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Formula- Normal distribution

m= mean N= total frequency X= no. of success = variance Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Properties of Normal distribution • It is a distribution of continuous variables • Takes values from -∞ to +∞ • It is symmetrical distribution; mean = median= mode • Assumes a bell shaped curve • It has two parameters ‘m’ and • First and third quartiles are equal distance from the median Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

• MD = 4/5 σ or 0.7979σ i. mean±2/3σ covers 50% of the observation ii. mean± σ covers 68.27% of the observation iii. mean±2σ covers 95.45% of the observation iv. mean±3σ covers 99.73% of the observation • All odd moments about mean=0 • Skewness is zero Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

•

Kurtosis is 3 its mesokutic, >3 its leptokutic, <3 its platykutic

â&#x20AC;˘If mean =0; SD =1 then the normal distribution is a standard normal distribution and its equation is

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Example Given a normal distribution of values for which the mean is 70 and the standard deviation is 4.5. â&#x20AC;˘ Find: a) the probability that a value is between 65 and 80, inclusive. b) the probability that a value is greater than or equal to 75. c) the probability that a value is less than 62. d) the 90th percentile for this distribution.

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,