The learning outcomes of this topic are: - recall the rules of simple probability - use key probability distributions (Binomial distribution, Poisson distribution, Exponential distribution, Normal distribution) - calculate z-scores This topic will cover: - Simple probability revision - Probability distributions - Standard scores (z-scores)

1. Topic 2:
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

4. 

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)

12. 

13. 

14. 

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

16. 

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)

18. 

19. 

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

21. 

22. 

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?