Section 4: Continuous Probability Distributions

Standard normal distribution

- mean of 0 - variance of 1

Tabulated values

- provide probabilities - in back of the book - Standard normal distribution

Normal Distribution: Need to know

1. E(X) = µ; V(X) = σ². 2. Area under curve = 1 3. Different means - shift curve up and down x-axis 4. Different variances - curve becomes more peaked or more squashed 5. Shorthand notation: X~N(µ, σ²).

Tools to use to find probabilities you want from Normal Tables

1. Symmetry → P(Z<-a) = P(Z>a) 2. Total area under curve is 1, total area under each half of curve is 0.5, i.e. P(Z<0)=P(Z>0)=0.5 3. Draw the curve, shade the area, break it up into areas you can find (differences or sums) 4. In general, if we cannot find the exact point of interest, choose the closest value available (beware of exceptions!) 5. If we are interested in a point so large that it is not in our tables, we consider the tail probability to be ≈0

The Normal Distribution formula

1. infinite support 2. need to know mean 3.need to know Variance

Continuous PDF's

1.) P(a<X<b) = area under a curve between a and b. the probability that x lies between a and b is the integral of the function f(x) dx. 2.) Probability that X will take any specific value is zero. 3.) A continuous random variable has a mean and a variance. - mean measuring the location of the distribution - variance measuring the spread of the distribution

Z-Score

= standardised value

Normal Distribution Shape

Bell shaped, symmetric around the mean approaches 0 asymptotically on either side of the mean

Continuous Data

Continuous data have an infinite number of possible values - smooth function used f(x) to describe the probabilities.

Probability Density Functions

F(x) must satisfy the following 1.) All probabilities have to be between 0-1 2.) Total area underneath the curve representing f(x) = 1 3.) f(x) curve never dips below the x-axis and never negative.

In General when standardising, must standardize a

Given X~N(μ,σ²), suppose we require P(X<a).

Standardising example graph

P(Z<1.5) = 0.9332 (from tables)

Uniform Distribution

Special sort of distribution for continuous data

Standardising Example

Suppose X~N(50, 100²). Find P(X<200).

Probabilities from the Normal Distribution

When Finding P(X>a)

Probabilities from the Normal Distribution Cont

When Finding P(a<X<b)

Standard Normal Distribution Notation

Z~N(0,1)

intergrate

finds the area under the curve

Standard normal tables

gives area underneath the standard normal curve

Discrete Data

has a countable number of possible values, Discrete probability Distributions can be put in a table

Standardising

have to convert any other normal distribution to the standard normal we can use existing tables.

Probability Density function

not important - Height of the curve or the value Important - area under the curve used to model a continuous random variable.

Continuous Random Variable

probability of an exact value is 0 (area of a line is 0)

PDF's

smooth f(x) function - probability that the variable lies in an interval (a,b) is the are under the curve from a to b. - values are not probabilities - area under the point is 0

Count Data

when there is a small amount of count data we treat it as ordinal. when there is a large amount of count data we treat it as continuous.