Lecture 02 Probability Distributions
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The document covers the topic of probability distributions, focusing on key aspects such as simple probability rules, different types of distributions (binomial, Poisson, exponential, and normal), and the calculation of standard scores (z-scores). It aims to equip students with the ability to recall probability rules and utilize various probability distributions effectively. Additionally, it includes explanations of sample spaces and events, and provides examples of discrete and continuous distributions.
Lecture 02 Probability Distributions
- 2. This topic will cover: ◦ Simple probability revision ◦ Probability distributions ◦ Standard scores (z-scores)
- 3. By the end of this topic students will be able to: ◦ recall the rules of simple probability ◦ use key probability distributions: Binomial distribution Poisson distribution Exponential distribution Normal distribution ◦ calculate z-scores
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- 5. ◦ Sample space Set of all possible outcomes ◦ Event One or more outcomes
- 6. E2 ◦ Mutually exclusive events events that cannot occur together P(E1 or E2) = P(E1) + P(E2) ◦ Non-mutually exclusive events P(E1 or E2) = P(E1) + P(E2) - P(E1 ∩ E2) E1 E1 E2E1∩ E2
- 8. ◦ Discrete Number of customers per hour Therefore seek model Probability Mass Functions that give P(X = x) Number Frequency Empirical Probability 0 10 0.0833 1 17 0.1417 2 42 0.3500 3 34 0.2833 4 12 0.1000 5 5 0.0417 120 1
- 9. ◦ Continuous height of customers therefore seek model probability density functions that lead to P(xl < X < xh) Height Frequency Empirical Probability 163 -165 1 0.005 166 -168 4 0.020 169 -171 14 0.070 172 -174 29 0.145 175 -177 44 0.220 178 -180 46 0.230 181 -183 35 0.175 184 -186 18 0.090 187 -189 7 0.035 190 -192 2 0.010 200 1
- 10. ◦ Sample space Set of all possible outcomes ◦ Event One or more outcomes ◦ Mean (of a random variable) 𝜇 = 𝑓𝑖 𝑥𝑖 𝑁 ⟶ 𝜇 = 𝑝𝑖 𝑥𝑖 ◦ Standard Deviation (of a random variable) 𝜎 = 𝑓𝑖 𝑥𝑖 − 𝜇 2 𝑁 ⟶ 𝜎 = 𝑝𝑖 𝑥𝑖 − 𝜇 2
- 11. ◦ A TRIAL has two possible outcomes P(success) = p, P(failure) = 1 - p Pass or fail training, medical treatment works or not, aeroplane engine works or not, meet SLA or not etc. ◦ Number of such trials, n, takes place 10 workers undergo training how many might pass? 1000 patients are treated, how many may recover? 4 working engines on aeroplane, how many will fail? ◦ Q ~ B(n, p)
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- 15. P(X ≥ 8) = 1- P(X ≤ 7) = 1 – 0.8327 = 0.1673 Probability distribution X ~ B(10,0.6) x P(X = x) P(X ≤ x) 0 0.0001 1 0.0016 2 0.0106 3 0.0425 4 0.1115 5 0.2007 6 0.2508 7 0.2150 8 0.1209 9 0.0403 10 0.0060 0.0001 0.0017 0.0123 0.0548 0.1662 0.3669 0.6177 0.8327 0.9536 0.9940 1.0000
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- 17. ◦ Rare event A in background of not A Large n and small p, np = l ◦ Probability of a number of independent, randomly occurring successes (or failures) within a specified interval Number of customers arriving at end of queue Number of print errors per area Number of machine breakdowns per year ◦ A ~ Po (l)
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- 20. Probability Distribution X ~ Po(6) x P(X = x) P(X ≤ x) 0 0.0025 1 0.0149 2 0.0446 3 0.0892 4 0.1339 5 0.1606 6 0.1606 7 0.1377 8 0.1033 P(x > 8) = 1 – 0.8472 = 0.1528 0.0025 0.0174 0.0620 0.1512 0.2851 0.4457 0.6063 0.7440 0.8472
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- 23. = 0.1474
- 24. s = 1 m = 0
- 25. ◦ Either tables or software can then give partial areas under the curve which indicate probabilities of particular values of z occurring. P(Z < z) P(Z > z)P(0 < Z < z)
- 26. -2.5 0 z
- 27. By the end of this topic students will be able to: ◦ recall the rules of simple probability ◦ use key probability distributions; Binomial distribution Poisson distribution Exponential distribution Normal distribution ◦ calculate z-scores
- 28. Any Questions?