PRELIM LEC 3 (3): PROBABILITY DISTRIBUTION Flashcards
It is a listing of observed / actual frequencies of all the outcomes of an experiment that occurred when experiment was done
Frequency Distribution
it is a listing of the probability of all the possible outcomes that could occur if the experiment was done
o It can be described as:
A diagram (Probability Tree)
A table
A mathematical formula
Probability Distribution
- Random variables can take only limited number of values.
- Ex. No. of heads in two tosses
Binomial Distribution
Poisson Distribution
Discrete Probability Distribution
- Random variables can take any value
- Ex. Height of students in the class Normal Distribution
Continuous Probability Distribution
There are certain phenomena in nature which can be identified as Bernoulli’s processes, in which:
o There is a fixed number of n trials carried out
o Each trial has only two possible outcomes say success or failure, true or false etc.
o Probability of occurrence of any outcome remains same over successive trials
o Trials are statistically independent
expresses the probability of one set of alternatives – success (p) and failure (q)
o P (X = x) = nrC pr qn-r (prob. of r successes in a trials) n = no. of trials undertaken r = no. of successes desired p = probability of success q = probability of failure
Binomial Distribution
When there is a large number of trials, but a small probability of success, binomial calculation becomes impractical
If ƛ = mean no. of occurence of an event per unit interval of time/space, then probability that it will occur exactly ‘x’ times is given by
P(x) = ƛx e-ƛ where e is napier constant and e = 2.7182 x!
POISSON DISTRIBUTION
Characteristics of Poisson Distribution
- It is a discrete distribution 2. Occurrences are statistically
- Mean no. of occurrences in a unit of time is proportional to size of unit
- It is always right skewed
- PD is a good approximation to BD when n > or = 20 and p < or = 0.05
Also called as Gaussian Distribution Develop by eighteenth century mathematician – astronomer Karl Gauss
It is symmetrical, unimodal (one peak)
The tails are asymptotic to horizontal axis.
X axis represents random variable like height, weight etc. Y axis represent its probability density function
The total area under the curve is 1 (or 100%)
Only two parameters are considered: Mean and Standard Deviation
Area under the curve tells the probability
o The mean ±1 standard deviation covers approximately 68% of the area under the curve
o The mean ±2 standard deviation covers approximately 95.5% of the area under the curve
o The mean ±3 standard deviation covers approximately 99.7% of the area under the curve
NORMAL DISTRIBUTION
distribution of values taken by the statistic in all possible samples of the same size from the same population
SAMPLING DISTRIBUTION
o There are three distinct distributions involved when we sample repeatedly and measure a variable of interest.
1. The population distribution gives the values of the variable for all the individuals in the population.
2. The distribution of sample data shows the values of the variable for all the individuals in the sample.
3. The sampling distribution shows the statistic values from all the possible samples of the same size from the population.
Population distribution vs. Sampling distributions
When we want information about the population proportion p of successes, we often take an SRS and use the sample proportion p ˆ to estimate the unknown parameter p. The sampling distribution of p ˆ describes how the statistic varies in all possible samples from the population
SAMPLE PROPORTION
o We have described the mean and standard deviation of the sampling distribution of the sample mean x but not its shape. That’s because the shape of the distribution of x depends on the shape of the population distribution
o In one important case, there is a simple relationship between the two distributions. If the population distribution is Normal, then so is the sampling distribution of x . This i s true no matter what the sample size is.
Sampling from Normal Population
o Most population distributions are not Normal. What is the shape of the sampling distribution of sample means when the population distribution isn’t Normal?
o It is a remarkable fact that as the sample size increases, the distribution of sample means changes its shape: it looks less like that of the population and more like a Normal distribution! When the sample is large enough, the distribution of sample means is very close to Normal, no matter what shape the population distribution has, as long as the population has a finite standard deviation.
o The central limit theorem (CLT) says that when is large, the sampling distribution of the sample mean x is approximately normal.
The Central Limit Theorem
