THEORY OF PROBABILITY MASLP
- 1. UNIT 4 STATISTICS: (Theory Of Probability- Principles And Properties Of Normal Distribution- Binomial Distribution- Interpretation Of Data Using The Normal Probability Curve- Causes Of Distribution- Deviations From The Normal Forms) SUBMITTED TO: SUBMITTED BY: DR. SATISK K. HIMANI BANSAL MVSCOSH MASLP 1ST YEAR
- 2. “ ” “Probability measures provide the decision maker in business and in government with the means for qualifying the uncertainties which affect his choice of appropriate actions.” -Morris Hamburg PROBABILITY
- 3. THEORIES OF PROBABILITY Addition Theorem • If two events are mutually exclusive and the probability of the one is P1, while that of the other is P2, the probability of either the one or the other occurring is the sum P1+P2 Multiplication Theorem • If two events are mutually independent and the probability of the one is P1, while that of the other is P2, the probability of the two events occurring simultaneously is the product of P1 and P2 Theorem of Conditional Probability • The probability that both of two dependent sub-events can occur is the product of the probability of the first sub-event and the probability of the second after the first sub-event has occurred Bayes’ Theorem • The concern for revising probabilities arises from a need to make better use of experimental information. This is referred to as Bayes’ Theorem
- 5. PROBABILITY DISTRIBUTION Normal Distribution Chi Square Distribution Binomial Distribution Poisson Distribution A list describing all the possible outcomes of a random variable along with their corresponding probabilities of occurrence In simple terms, it shows the likelihood of different possible outcomes of a variable
- 8. STANDARD NORMAL DISTRIBUTION & STANDARD SCORES Z- distribution/Standard normal distribution: a special case of the normal distribution where the mean is zero and the standard deviation is 1 A value on the standard normal distribution is known as a standard score or a Z-score Z- score: represents the number of standard deviations above or below the mean that a specific observation falls
- 9. STANDARDIZATION: HOW TO CALCULATE Z- SCORE Standardization: to take observations drawn from normally distributed populations that have different means and standard deviations and place them on a standard scale. To standardize our data, we need to convert the raw measurements into Z-scores. Here, X represents the raw value of the measurement of interest. Mu and sigma represent the parameters for the population from which the observation was drawn.
- 10. FINDING AREA UNDER THE NORMAL DISTRIBUTION CURVE The proportion of the area that falls under the curve between two points on a probability distribution plot indicates the probability that a value will fall within that interval We can calculate areas by looking up Z-scores in a Standard Normal Distribution Table We can transform the values from any normal distribution into Z-scores, and then use a table of standard scores to calculate probabilities
- 11. IMPORTANCE OF NORMAL DISTRIBUTION CURVE
- 12. CENTRAL LIMIT THEOREM For a sufficiently large sample size, the sampling distribution of the mean for a variable will approximate a normal distribution regardless of that variable’s distribution in the population As the sample size increases, the sampling distribution converges on a normal distribution where the mean equals the population mean, and the standard deviation equals σ/√n. Where: σ = the population standard deviation; n = the sample size In a population, the values of a variable can follow different probability distributions. These distributions can range from normal, left-skewed, right-skewed, and uniform among others.
- 15. QUESTIONS ASKED IN PREVIOUS YEARS 1. The incidence of occupational disease in an industry is such that the workmen have a 20% chance of suffering from it. What is the probability that out of six workmen 4 or more will contact the disease? (8 marks, 2019) 2. Write a short note on Binomial Distribution. (4 marks, 2018) 3. Narrate the properties and importance of Normal Distribution. Given the mean and SD of serum iron values for healthy men are 120 and 150 micrograms. In a sample of 500 cases, assuming normality, estimate the number of individuals with serum iron values less than 135 microgram. Also, find the probability that people will get serum value from 90 to 135 microgram? (16 marks, 2018) 4. Explain the properties and importance of the normal distribution. (8 marks, 2017) 5. A study on the FBS levels of 100 diabetics gave a mean of 156 mg/dl and a SD of 12 mg/dl. Assuming a normal distribution, what is the probability that any given individual will have an FBS level between 132 and 168 mg/dl? Estimate the number of subjects with FBS levels less than 168 mg/dl. (8 marks, 2017) 6. Explain the properties of Normal distribution and Binomial distribution. (12 marks, 2016) 7. Suppose the average length of stay in a chronic disease hospital of a certain type of the patient is 60 days with a standard deviation of 15. By assuming a normal distribution, find the probability that a randomly selected patient will have a length of stay between 30 and 60 days. (4 marks, 2016) 8. What is the importance of normal distribution? State its properties. (8 marks, 2013) 9. Short note on normal probability distribution. (8 marks, 2012, 2011)
- 16. REFERENCES 1. Statistics, Gupta B.N. (1991) 2. https://www.britannica.com/science/probability-theory/An-alternative-interpretation-of-probability 3. https://www.cuemath.com/data/probability-theory/ 4. https://egyankosh.ac.in/bitstream/123456789/23467/1/Unit-4.pdf 5. https://statisticsbyjim.com/basics/normal-distribution/ 6. https://statisticsbyjim.com/basics/central-limit-theorem/ 7. https://www.wallstreetmojo.com/law-of-large-numbers/