STATISTIC & INFORMATION THEORY (CSNB134) MODULE 7A PROBABILITY DISTRIBUTIONS FOR RANDOM VARIABLES (BINOMIAL DISTRIBUTION)
Overview • In Module 7, we will learn three types of distributions for random variables, which are: - Binomial distribution - Module 7A - Poisson distribution - Module 7B - Normal distribution - Module 7C • This is a Sub-Module 7A, which includes lecture slides on Binomial Distribution.
The Binomial Random Variable • Example of a binomial random variable: • coin-tossing experiment - P(Head) = ½. • Toss a fair coin n = 3 times, x = number of heads. • We have seen in Module 6 that the probability distribution for the coin-tossing experiment, where a coin is toss for 3 times and x = number of heads is as follows.
The Binomial Random Variable (cont.) • Many situations in real life resemble the coin toss, but with a coin that might not be fair, i.e. P(H) 1/2. • Example: A lecturer samples 10 people and counts the number who of students who are from KL. • The analogy to coin tossing is as follows: • Coin: • Head / H: • Tail / T: • Number of tosses: • P(H): Student From KL Not from KL n = 10 P(From KL) = proportion of students in the population who are from KL.
The Binomial Random Variable (cont.) • Other examples: • Proportion of COIT students who are female. • Proportion of Malaysian population who watch Akademi Fantasia. • Proportion of people in Bangi who subscribe to ASTRO. Female/Male P(F) = .55 / P(M) = .45 Watch/Don’t Watch P(W) = .35 / P(DW) = .65 P(S) = .85 / P(DS) = .15 Subscribe/Don’t Subscribe All exhibit the common characteristic of Binomial experiment!
Characteristic of a Binomial Experiment • The experiment consists of nidentical trials. • Each trial results in one of two outcomes, success (S) or failure (F) i.e: mutually exclusive. • The probability of success on a single trial is pand remains constant from trial to trial. The probability of failure is q = 1 – p. • The trials are independent (Note = please refer notes on Module 5 on ‘Independent Event’). • We are interested in x, the number of successes in n trials.
Binomial or Not? • Example: Select two people from the U.S. population (say 300 million), and suppose that 15% (45 million) of the population has the Alzheimer’s gene. • For the first person: p = P(gene) = 45,000,000/300,000,000 = .15 • For the second person: • p = P(gene) = 44,999,999/299,999,999 .15, even though one person has been removed from the population. • We assume that the removal of one person from the population have negligible effect.
The Binomial Probability Distribution • For a binomial experiment with n trials and probability pof success on a given trial, the probability of ksuccesses in n trials is: • And the measures of center and spread are:
Exercise 1 • A fair coin is flipped 6 times. (1) What is the probability of obtaining exactly 3 heads? • For this problem, N =6, k=3, and p = .5, thus: • (2) Determine the probability of obtaining 3 or more successes. • P(x >=3) = P(3) + P(4) + P(5) + P(6) = pls find this out…..
Exercise 2 A rifle shooter hits a target 80% of the time. He fires five shots at the target. Assuming x = number of hits, what is the mean and standard deviation for x?
Exercise 2 (cont.) • Would it be unusual to find that none of the shots hit the target? • The value x = 0 lies • more than 4 standard deviations below the mean. Very unusual!! (Note: Refer to z-score in Module 3)
p = x = n = success = Exercise 2 (cont.) What is the probability that exactly 3 shots hit the target? 5 hit .8 # of hits
Exercise 2 (cont.) What is the probability that more than 3 shots hit the target?
Example 2 (Cont.) For the same rifle shooter, if he fires 20 shots at the target. What is the probability that more than 5 shots hit the target?
Effect of p Two binomial distributions are shown below. Here, π is the same as p Notice that for p = .5, the distribution is symmetric whereas for p = .3, the distribution is skewed right.
Probability Tables • We can use the binomial probability tables to find probabilities for selected binomial distributions. • They can either be • Probability tables for each x = k or • Cumulative probability tables
Probability Tables (cont.) Probability table for each x = k (part of)
Probability Tables (cont.) Cumulative Probability table(part of) (for n= 5) Probability table for each x = k (for n = 5) P(x<=3) = 0.263 (from the cumulative table) P(x<=3) = 0.000 + 0.006 + 0.051 + 0.205 (from the x=k table) = 0.262
Cumulative Probability Tables We can use the cumulative probability tables to find probabilities for selected binomial distributions. • Find the table for the correct value of n. • Find the column for the correct value of p. • The row marked “k” gives the cumulative probability, P(x k) = P(x = 0) +…+ P(x = k)
Exercise 3 A rifle shooter hits a target 80% of the time. He fires five shots at the target. What is the probability that exactly 3 shots hit the target? • Find the table for the correct value of n. • Find the column for the correct value of p. • Row “k” gives the cumulative probability, P(x k) = P(x = 0) +…+ P(x = k)
Exercise 3 (cont.) P(x = 3) = P(x 3) – P(x 2) = .263 - .058 = .205 Check from formula: P(x = 3) = .2048
Exercise 2 (cont.) What is the probability that more than 3 shots hit the target? P(x > 3) = 1 - P(x 3) = 1 - .263 = .737 Check from formula: P(x > 3) = .7373
STATISTIC & INFORMATION THEORY (CSNB134) PROBABILITY DISTRIBUTIONS OF RANDOM VARIABLES (BINOMIAL DISTRIBUTIONS) --END--