Example

The probability of choosing statistics is 0.16. If 20 students are randomly selected, what is the probability that

• at least 5 would choose the subject?
• exactly 2 would choose the subject?
• none will choose?

Function

\(P(x) = ^nC_x \space p^x(1-p)^{n-x}\)

\(1-p=q\)

• \(p \to\) probability of success
• \(q \to\) probability of failure
• \(n \to\) number of Bernoulli trials
• \(x \to\) no. of success

Bernoulli Trial

• Introduced by James Bernoulli
• Independent trials with dichotomous outcomes (success/failure)
• Coin toss, disease detection

Bernoulli Distribution

\(p^xq^{1-x}\)

If n = 1 in binomial distribution

Properties of Binomial Distribution

• Mean: \(np\)
• Variance: \(npq\)
• Skewness, \(\beta_1 = \frac{(q-p)^2}{npq}\)
• Kurtosis, \(\beta_2 = 3 + \frac{1-6pq}{npq}\)

Derivation of pmf

\(P(SF SF \cdots) = P(S) \times P(F) \cdots\)

• \((p \times p \cdots \times p)(\times q \times q \cdots \times q)\)
• \(p^xq^{n-x}\)
• \(^nC_x \space p^xq^{n-x}\) (any x combination from n)
• 2 items from a, b, c \(\to ab, ac, bc \to \space ^3C_2\)

Derivation of Mean

\(\displaystyle E(X) = \sum_{x=0}^n x\cdot p(x)\) (think: why zero?)

• \(0 \cdot p(0) + \cdots\)
• \(np\{q^{n-1}+n(n-1)pq^{n-2}+\cdots+p^{n-1}\}\)
• \(np(q+p)^{n-1} [binomial \ theorem]\)
• \(np\) [p+q=1]

Uses of Binomial Distribution

• Quality control
• Test of fit

Binomial to Poisson

See here

Binomial to Noraml

If \(n \to \infty\) and \(np\ge 5, nq \ge 5\)

Problem 01

The mean and sd of a binomial distribution are 40 and 6. Find

1. \(n,p,q\)
2. \(P(x)\)
3. \(P(x=0), P(x\ge2), P(x\lt3)\)

• \(np = 40\)
• \(\sqrt{npq} = 6\)
• \(np(1-p)=36 \to 40(1-p)=36 \to 1-p=.9\)

Problem 02

An unbiased coin is tossed 10 times. Find the probability of

1. 3 heads
2. at least 2 heads
3. at most 1 head

Problem 03

The probability that it rains on a particular day is 0.2. If 10 days are observed, what is the probability that it would rain on

1. all 10 days
2. no days
3. at least 2 days
4. on exactly four days

Problem 04

In El Clasico, the probability that Barcelona wins is 0.39 and that Real Madrid wins is 0.42. In the next 5 matches, what is the probability that

1. Barcelona wins all the matches?
2. Real wins all the matches
3. Barcelona wins at least 2 matches
4. 2 matches are drawn
5. more than 3 matches are drawn

• \(p_1=\)Barca wins, \(p_2 =\) Real wins, \(p_3 =\)Match drawn