Lecture 3 (PROBABILITY DISTRIBUTIONS)

Question 1

RANDOM VARIABLE

Answer 1

A variable which contains the outcomes of a chance experiment.

Question 2

DISCRETE DISTRIBUTIONS

Answer 2

Constructed from discrete (individually distinct) random variables.

Question 3

CONTINUOUS DISTRIBUTIONS

Answer 3

Based on continuous random variables.

Question 4

DISCRETE RANDOM VARIABLES

Answer 4

The set of all possible values is at most a finite or a countable infinite number of possible values.
E.g. Number of new subscribers to a magazine
Number of absent employees on a given day

Question 5

CONTINUOUS RANDOM VARIABLE

Answer 5

Takes all possible values in some interval.
Percentage of labour force that is unemployed
A person’s weight

Question 6

Describing a discrete distribution

Answer 6

A discrete distribution can be described by constructing a graph of the distribution.
Measures of central tendency and variability can be applied to discrete distributions.
Discrete values of outcomes are used to represent themselves.

Question 7

MEAN OF DISCRETE DISTRIBUTION

Answer 7

Long run average
If process is repeated long enough, the average of the outcomes will approach the long run average (mean).
Requires the process to eventually have a number which is the product of many processes.

u = E(x) = Σ[X * P(X)]
E is the long run average
X = an outcome
P = Probability of X

Question 8

Variance of discrete distrbution

Answer 8

Weighted average of squared deviations about the mean.

o^2 Σ(X-u)^2*P(X)

Question 9

BINOMIAL DISTRIBUTION

Answer 9

Experiment involves n identical trials, where n is fixed before the trials are conducted.
Each trial has exactly two possible outcomes: success and failure.
Each trial is independent of the previous trials:
p is the probability of success on any one trial
q = 1-p is the probability of a failure on any one trial.
P and q are constant throughout the experiment.
X is the number of successes in the n trials: p = X/n (relative frequency probability)
P and n are known as the parameters of a binomial distribution

APPLICATIONS:
Sampling with replacement
Sampling without replacement - n<5%N

Question 10

Mean and Std Dev of Binomial Distribution

Answer 10

Mean value:
u = n*p
Variance and Std Dev:
o^2 =n*p*q
o = sqrt  of o^2 = sqrt of n*p*q

Question 11

CONTINUOUS DISTRIBUTIONS

Answer 11

Constructed from continuous random variables in which values are taken for every point over a given interval.
The probabilities of outcomes occurring between particular points are determined by calculating the area under the curve between these points.

Question 12

CHARACTERISTICS OF THE NORMAL DISTRIBUTION

Answer 12

Continuous distribution - line does not break
Symmetrical distribution - each half is a mirror of the other half.
Asymptotic to the horizontal axis - it does not touch the x axis and goes on forever.
Unimodal - means the values mound up in only one portion of the graph
Area of the curve = 1; total of all probabilities = 1

Question 13

What is normal distribution characterised by?

Answer 13

The mean and the standard deviation
Every unique pair of u and o values define a different normal distribution. Changes in these give a different distribution.

Question 14

Z-SCORE

Answer 14

The conversion formula for any x value of a given normal distribution.
z = (x-u)/o
Gives the number of standard deviations that value of x is above or below the mean
if x value is less than the mean, the z score is negative
If x value is greater than the mean, the z score is positive.
Can be used to find probabilities for any normal curve that has been converted to z scores
Mean of 0 and a std dev of 1