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Introduction to Probability : Binomial & Normal Distribution

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Introduction to Probability : Binomial & Normal Distribution

Dr. Marvin Reid

- Define probability and its importance in statistical theory
- Describe the associative and multiplicative rules for joint probability under statistical independence
- Describe the properties and uses of 2 probability distribution functions the normal & binomial distributions.

- Populations
- samples drawn from these populations

- Methods used to summarize the data obtained from these samples
- The relation between sample and population is uncertain. Thus to make inferences about our data we need to set up mathematical models which capture this uncertainty
- The foundation of statistical models is probability theory

- Relative frequency
- Degree of belief
- Probability reasons from the population to the sample
- Probability lies between 0 and 1

- Trial (Experiment) – any process generates a set of results
- Outcome – the result of carrying out the trial
- Event – one or more outcomes
- Marginal probability –one event occurs

- Mutually exclusive if the events cannot occur simultaneously
- Independent if the occurrence of an event does not influence the probability of another event occurring

Start

No

Addition Rule

P(A or B)=P(A)+P(B)-P(AB)

Are events mutually exclusive

yes

Addition Rule

P(A or B)=P(A)+P(B)

Joint Probability

P(AB)=P(A) x P(B)

Are events statistically independent

yes

Marginal Probability

P(A)

- Many statistical methods use probability distribution
- Probability distribution is used to calculate the theoretical probability of different values occurring
- Normal distribution – continuous data
- Binomial distribution- discrete data

- Extends from –infinity to +infinity
- Height=probability density
- Area under curve=1
- Unimodal
- mean=median=mode

y1;µ=50;σ=5

y2;µ=50;σ=10

Completely described by the mean & sd

Any normally distributed variable can be related to the standard normal distribution whose mean is zero

and standard deviation is 1. This can be done by performing the following calculation

Z is the distance along the x axis in sd units

95%

- Used to calculate probability of values being within specified range eg 95%CI= m ± (1.96 x se)
- Used to test inferences about the difference between a single mean and a hypothesized value and the difference between two means

- Describes discrete data resulting from experiments called Bernoulli process
- Each experiment (trial) has only 2 possible outcomes.
- The probability of the outcome of any trial remains fixed over time.
- The trials are statistically independent.

- Example = Toss of fair Coin

- p=probability of success
- q=(1-p)=probability of failure
- r=number of successes desired
- n=number of trials undertaken

Binomial Formula

- A couple each with sickle trait have 4 children. What is the probability that two children will have sickle cell disease.
- P(SS)=0.25, q(Non-SS)=0.75, n=4.

- When p is small the binomial distribution is skewed to the right
- As p increases the skewness is less noticeable
- When p=.5 the binomial distribution is symmetrical
- When p >0.5 the distribution is skewed to the left
- As n increases binomial distribution approximates the normal distribution (np and nq>5)

P=0.1

P=0.5

P=0.7

P=0.4

Mean of a Binomial Distribution

Standard Deviation of a Binomial Distribution

Standard error of the proportion

- Used to calculate probability of values being within specified range eg CI
- Used to test inferences about the difference between a single proportion and a hypothesized value and the difference between two proportions

Z is the appropriate percentage point of the normal distribution

- Dr. McGaw-Binns surveyed 150 medical students and found that 42% of them had a sedentary lifestyle
- A) Estimate the standard error of the proportion
- B) Construct a 95% confidence interval for the true proportion of students who had a sedentary lifestyle

N=150, p=0.42

- The stata command
- cii number probability
- cii 150 0.42

-- Binomial Exact --

Variable | Obs Mean Std. Err. [95% Conf. Interval]

-------------+-------------------------------------------------------------

| 150 .42 .0402989 .3399811 .503244

- p1 and p2 are the proportions
- se=standard error
- p=overall proportion based on the two sample proportions

Compare calculated z with the appropriate percentage point Zαof

the normal distribution appropriate

- Define probability and its importance in statistical theory
- Describe the associative and multiplicative rules for joint probability under statistical independence
- Describe the properties and uses of 2 probability distribution functions the normal & binomial distributions.