4.3 - Binomial Random Variable

Familiar Population

{A, B, C, D, E}

Interest - number of vowels in population
Sampling - i.i.d. with $n = 2$

Computations

$N = 25$
A - number of vowels

A	$P (A)$	$A \cdot P (A)$	$(A - μ)^{2}$	$(A - μ)^{2} \cdot P (A)$
0	$\frac{9}{25}$	0	0.64	0.2304
1	$\frac{12}{25}$	$\frac{12}{25}$	0.04	0.0192
2	$\frac{4}{25}$	$\frac{8}{25}$	1.44	0.2304

$μ = 0 + \frac{12}{25} + \frac{8}{25} = \frac{20}{25} = 0.8$
$σ^{2} = 0.48$
$σ = 0.6928$

Conducting Inferential Statistics

Population - ${A, B, C, D, E}$
Interest - number of vowels
Sampling method - i.i.d. with $n = 3$

# of vowels	P(# of vowels)
0	$P (CCC) = P (C) \cdot P (C) \cdot P (C) = {\frac{3}{5}}^{2} = \frac{27}{125}$
1	$P (VCC or CVC or CCV) = P (VCC) + P (CVC) + P (CCV) = \frac{56}{125}$
2	$P (VVC or VCV or VCC) = \frac{36}{125}$
3	$P (VVV) = \frac{8}{125}$

Expected value - $μ = \sum (P (A_{i}) \cdot A) = 1.2$
Variance - $σ^{2} = \sum ((μ - x_{i})^{2} \cdot P (A_{i})) = 0.72$

Binomial Random Variable

Random experiment
Outcomes classified as success S or failure F
Probability of success when running the experiment does not change $$P(S) = p$$
Probability of failure does not change as well $$P(F) = q = 1 - p$$
The random experiment is to be run for a fixed number of trials $n$
The trials are identical for each run
The probabilities are independent across runs
The random variable is constructed by counting the number of successes in the $n$ trials

IMPORTANT

This does not work for SRS sampling, as for each of the choosings we'd have 1 less population, but works in i.i.d.!!!

On Inferential Statistics

Population - ${A, B, C, D, E}$
Interest - number of vowels in population
Sampling method - i.i.d. with $n = 4$

Explain why this example can be understoodas a binomial random variable
we're choosing one person at a time, and putting them back, so the chance of choosing a vowel is always the same

Possible values of random variable

Y	P(Y)
0	$P (FFFF) = {\frac{3}{5}}^{4}$
1	$P (SFFF or FSFF or FFSF or FFFS) = 4 \cdot {\frac{3}{5}}^{3} \cdot \frac{2}{5}$
2	$P (SSFF or SFSF or SFFS or FSSF or FSFS or FFSS) = 6 \cdot {\frac{2}{5}}^{2} \cdot {\frac{3}{5}}^{2}$
3	$P (SSSF \dots) = 4 \cdot \frac{3}{5} \cdot {\frac{2}{5}}^{3}$
4	$P (S S S S) = {\frac{2}{5}}^{4}$

For binomial random variable, the possible values of random variable will be always

1 \dots N + 1

To create this table automatically, we can see this

BRV	P()
0	$q^{n}$
...
i	$(\binom{n}{i}) \cdot p^{i} \cdot q^{n - i}$
...
n	$p^{n}$

Probability Distribution Function

To find the probability of binomial random variable $X$ with $n$ trials and probability of success equal to $p$ , we can use a formula: $$ P(X) = \binom{n}{X} \cdot p^X \cdot q^{n-X} $$

P (X \leq x) = \sum_{i = 0}^{n} (\binom{n}{i}) \cdot p^{i} \cdot q^{n - i}

Additional Formulas

Those only work for binomial random variables

μ = n \cdot p

σ^{2} = n \cdot p \cdot q

σ = \sqrt{n \cdot p \cdot q}

Exercise

Construct a binomial random variable with a 100 possible values such that the expected values is 33
For 100 possible values, $n = 99$
$μ = 33 = n \cdot p$
$33 = 99 \cdot p \to p = \frac{1}{3}$