1. 26134 Business Statistics Autumn 2017
Tutorial 5: Continuous Probability Distributions B MathFin (Hons) M Stat (UNSW) PhD (UTS) mahritaharahap.wordpress.com/ teaching-areas
business.uts.edu.au

2. Last week revision

3. What is a random variable?
A random variable, usually written X, is a variable whose possible values are numerical outcomes of a random phenomenon. There are two types of random variables, discrete and continuous.

5. Binomial Distribution
𝑓 𝑥 =𝑃 𝑋=𝑥 =nCx pxq(n-x)
where: f(x) = probability of x successes in n independent trials
p = probability of success
q = 1-p = probability of failure
𝑥 = number of successful trials n-x = number of failed trials

6. 𝑓 𝑥 =𝑃 𝑋=𝑥 = 𝜆 𝑥 𝑒 −𝜆 𝑥! Poisson Distribution where:
𝑓 𝑥 =𝑃 𝑋=𝑥 = 𝜆 𝑥 𝑒 −𝜆 𝑥!
where:
f(x) = probability of x occurrences in a specified interval
𝜆 = mean number of events in a specified interval

7. Hypergeometric Distribution
𝑓 𝑥 =𝑃 𝑋=𝑥 = rCxN−rCn−x NCn
where:
f(x) = probability of x successes in n dependent trials
n = number of trials
N = number of elements in the population
r = number of elements in the population labelled success
x= number of elements in the sample labelled success

8. Cumulative Probability
Discrete Distributions
P(X<=x)≠P(X<x)
P(X=x) ≠ 0
i.e. P(X<=2) ≠ P(X<2)
Continuous Distributions
P(X<=x)=P(X<x)
P(X=x) ≈ 0
i.e. P(X<=2)=P(X<2)

9. Continuous Distributions

10. Activity 1: Uniform Distribution

12. Activity 2: Normal Distribution
X=rate of return on a proposed investment
𝑋~𝑁(μ=0.30 ,σ=0.10)
𝑃 𝑋≤$0.23 =𝑃(𝑍≤ 𝑥−μ σ )
= 𝑃(𝑍≤ 0.23− )
= 𝑃(𝑍≤−0.70) =
𝑍~𝑁(μ=0 ,σ=1)

15. We need to do this to standardise the distribution so we can find the probabilities using the tables.
To find probabilities under the normal distribution, random variable X must be converted to random variable Z that follows a standard normal distribution denoted as Z~N(μ=0,σ=1). We need to do this to standardise the distribution so we can find the probabilities using the tables.

16. Why do we standardise the normal distribution to find probabilities?
When we are given a problem about a random variable X~N(μ,σ) that follows a normal distribution, to find cumulative probabilities P[X<x] we would have to calculate a complex definite integral.
To do this computation easier, we have standard practices in statsistics where we can convert the normal random variable X~N(μ,σ) into a standard normal random variable Z~N(μ=0,σ=1) (by computing the associated z-score, then we can just look up the tables to find certain probabilities): @

17. @ Mahrita.Harahap@uts.edu.au
Calculating Probabilities using normal distribution applying the complement rule and/or symmetry rule and/or interval rule
Complement Rule P(Z>z)=1-P(Z<z)
Symmetry Rule P(Z<-z)=P(Z>z)
Interval Rule P(-z<Z<z)=P(Z<z)-P(Z<-z) @

19. Activity 3: Poisson and Exponential Distributions
X = time of visits
𝜆 =1/ μ = mean number of events per unit of time = two clients to talk to on average per 30mins = one client to talk to on average per 15mins =1/15 clients on average per minute = clients per minute

23. Activity 3: Poisson and Exponential Distributions
X = time of visits
𝜆 =1/ μ = mean number of events per unit of time = two clients to talk to on average per 30mins = one client to talk to on average per 15mins =1/15 clients on average per minute = clients per minute
P(X<x) = 1 - e-λx
P(X<10)=1-e *10=
We would use this for Poisson:
𝜆 = 2 clients to talk to on average per 30mins = 2/3 client to talk to on average per 10mins

24. Revision

25. In statistics we usually want to analyse a population parameter but collecting data for the whole population is usually impractical, expensive and unavailable. That is why we collect samples from the population (sampling) and make conclusions about the population parameters using the statistics of the sample (inference) with some level of confidence (level of significance).
In statistics we usually want to analyse a population parameter but collecting data for the whole population is usually impractical, expensive and unavailable. That is why we collect samples from the population (sampling) and make conclusions about the population parameters using the statistics of the sample (inference) with some level of confidence (level of significance).
Week 1

26. Types of data – Graphical displays
Categorical
(divides the cases into groups/categories)
Nominal (no order) e.g. Nationality, Gender, Month
Ordinal (order) e.g. Satisfaction Level or level of education
Quantitative
(measures a numerical quantity for each case)
Discrete takes whole number values e.g. number of birds in a tree, shoe size
Continuous e.g. height, temp
Bar Chart
Pie Chart
Boxplot
Histogram
Week 1

27. Presenting Data Graphically
Univariate
Categorical Data
Quantitative Data
Bar Charts to depict frequencies
Pie Charts to depict proportions
Histogram to look at the distribution
Boxplot graphical summary statistics

28. Displaying categorical data
Bar Graph
Pie Graph
Good for comparing the number of individuals in different groups.
Good for looking at parts as a whole.
Good for depicting frequencies.
Good for depicting proportions
Week 1

29. Displaying quantitative data
Histogram
Boxplot
The graphs on the next slide correspond to the recorded change in pulse rate of 92 students after half of the students ran on the spot for a minute and half of the students rested for a minute.
Can you identify these two groups in the boxplot and histogram?
Good for getting an overall ‘picture’
of the data
Good for finding unusual observations and
looking at the symmetry of the data
Week 1

30. Presenting Data Graphically
Bivariate
Categorical x Categorical
Categorical x Quantitative
Quantitative x Quantitative
Multiple Bar Charts to compare frequencies
Crosstabs
to depict frequencies, row and column proportions
Multiple Comparison Boxplots
Scatterplots relationship between two numerical variables

31. Types of data – Measures of central tendency
Categorical
Nominal (no order) e.g. Nationality, Gender
Ordinal (order) e.g. S,M,L or level of education
Quantitative
Discrete takes whole number values e.g. number of birds in a tree
Continuous e.g. height
Mode
Median
or Mode
Mean, Median or Mode
Mean or Median or Mode
Week 1

32. Types of data – Measures of spread
Categorical
Nominal (no order) e.g. Nationality, Gender
Ordinal (order) e.g. S,M,L or level of education
Quantitative
Discrete takes whole number values e.g. number of birds in a tree
Continuous e.g. height
None
IQR
Range, IQR SD CV
Range, IQR SD CV
Week 1

33. Measures of dispersion
The coefficient of variance is used when you are comparing the variability between groups with different means.
It is defined as the ratio of the standard deviation to the mean, expressed as a percentage.
Dividing the standard deviation by the mean standardises the measure of variability so it is suitable for comparison.
𝐶𝑉= 𝑠 𝑥 ∗100
Week 1

34. @ Dr. Sonika Singh, BSTATS, UTS
Summary: Probability
Marginal Probability
Union Probability
Joint Probability
Conditional Probability
Independent Probability U
@ Dr. Sonika Singh, BSTATS, UTS

35. Conditional Probability
𝑃 𝐴 𝐵 = 𝑃(𝐴⋂ 𝐵) 𝑃(𝐵)

36. Independent Events
𝑃 𝐴⋂𝐵 =𝑃 𝐴 ∗𝑃(𝐵)

37. Contingency Analysis – Test for Independence (Chi-Square test)
Critical Value=CHIINV(0.05,df)
Conclusion:
If χ2>critical value we reject Ho. We conclude that at the 5% level of significance we have enough evidence to show that the two variables are not independent.
If χ2<critical value we do not reject Ho. We conclude that at the 5% level of significance we do not have enough evidence to show that the two variables are not independent.

38. Discrete Distributions
Binomial Distributions
Week 1

39. Continuous Distributions
Uniform X~Uniform(a,b)
Normal X~Normal(μ,σ)
Exponential X~Exp(λ)
Week 1

40. SEE YOU ALL NEXT WEEK!