Binomial Distribution

Discrete Random Variable 

: X~ Bin(n, p) = # of success in n Bernoulli trials each with success probability p.

$\bigstar P(X=k)=\left ( \frac{n}{k} \right )p^k(1-p)^{n-k}$, x=0,1,...,n
$\bigstar$ E(X)=np, Var(X)=np(1-p)=npq (where q=1-p) 


Finding Variance w/o formula Example) 

Let Y~be binomial (15, $\frac {1}{3}$). Evaluate Var(Y).  

(We know variance of binomial distribution is npq. However what if we don't know this formula?)

Solution??!!

Example) Mathematical Statistics and Data Analysis 3ED Chapter8. Q31.

George spins a coin three times and observed no heads. He then gives the coin to Hilary. She spins it until the first head occurs, and ends up spinning it four times total. Let $\theta $ denote the probability the coin comes up heads. 

a) What is the likelihood of $\theta $?

b) What is the MLE of $\theta $?

Solution??!!

Hypothesis Testing 

Example) Mathematical Statistics and Data Analysis 3ED, Chapter 9. Q1. 

A coin is thrown independently 10 times to test the hypothesis that the probability of head is $\frac {1}{2}$ . $H_{1}: p\neq \frac {1}{2}$. The test rejects if either 0 or 10 heads are observed. 

a) What's the significance level of test?

b) If in fact, the probability of head is 0.1. What's the power of the test? 

Solution??!!

Example) 

An experimenter has prepared a drug dosage level that she claims will induce sleep for 80% of people suffering from insomnia. After examining the dosage, we feel that her claims regading the effectiveness of the dosage are inflated. In an attempt to disprove her claim, we administer her prescribed dosage to 20 insominiacs and we observe Y, the number for whom the drug tdse induces sleep. The rejection region was found to be {Y $\leq$ 12}. 

(a)  $H_{0}$? $H_{1}$?

(b) In terms of this problem, what's a Type I error? 

(C) find $\alpha$ 

(d) In terms of this problem, what's a Type II error? 

(e) Find  $\beta$ when p=0.6

(f) Find $\beta$ when p=0.4

Solution??!!