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(a) For the class 5 – 9:
Lower class boundary is 4.5 Upper class boundary is 9.5 Class width = 5 Class midpoint - "x" = 7
Frequency density = Frequency / Class width
The sample space is every single possible outcome, while an event is a set of possible outcomes.
A Venn diagram shows the sample space, and a Tree diagram shows the events.
Discrete Random Variables
Discrete Random Variables are ones which can only take certain values, and not the values in between. A probability function describes the probability of each outcome, and a probability distribution is a table of all the outcomes along with their probabilities. A cumulative distribution function is in the form F(x), and is the probability of all the values of outcomes up to and including x.
The sum of all probabilities must equal 1.
Discrete Uniform Distribution
Not necessary, but saves time in an exam. A discrete uniform distribution is when all outcomes have an equal probability of occurring (e.g. a die roll). For a discrete uniform distribution:
Continuous Random Variables
The Normal Distribution
The probability distribution of a continuous random variable is represented by a curve; the area under the curve in a given interval gives the probability of a value lying in that interval.
If X is normally distributed with mean µ and standard deviation σ, then X~N(µ,σ^2)
If Z is a continuous random variable, where Z~N(0,1) then Φ(z) = P(Z<z)
The variable Z=(X-μ)/σ is the standard normal variable corresponding to X.
The percentage points table shows, for a probability p, the value of z such that P(Z<z) = p
2 variables are positively correlated if one increases with the other, and negatively correlated if one decreases as the other increases. The variables are usually plotted on a scatter diagram. Correlation is measured by the Product Moment Correlation Coefficient (PMCC), r, where:
Estimating and Samples