Assumptions of Binomial Distribution
- Trials are independent
- p, q remain constant for each trial
- Outcomes are dichotomous
- X counts number of success
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 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
- \(n,p,q\)
- \(P(x)\)
- \(P(x=0), P(x\ge2), P(x\lt3)\)
- \(np = 40\)
- \(\sqrt{npq} = 6\)
- \(np(1-p)=36 \to 40(1-p)=36 \to 1-p=.9\)
Problem 02
An unbiased coin is tossed 10 times. Find the probability of
- 3 heads
- at least 2 heads
- 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
- all 10 days
- no days
- at least 2 days
- 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
- Barcelona wins all the matches?
- Real wins all the matches
- Barcelona wins at least 2 matches
- 2 matches are drawn
- more than 3 matches are drawn
- \(p_1=\)Barca wins, \(p_2 =\) Real wins, \(p_3 =\)Match drawn
