Normal Distribution Introduction

Probability Density Functions

8.3 Probability Density Functions… Unlike a discrete random variable which we studied in Chapter 7, a continuous random variable is one that can assume an uncountable number of values.  We cannot list the possible values because there is an infinite number of them.  Because there is an infinite number of values, the probability of each individual value is virtually 0.

8.4 Point Probabilities are Zero  Because there is an infinite number of values, the probability of each individual value is virtually 0. Thus, we can determine the probability of a range of values only. E.g. with a discrete random variable like tossing a die, it is meaningful to talk about P(X=5), say. In a continuous setting (e.g. with time as a random variable), the probability the random variable of interest, say task length, takes exactly 5 minutes is infinitesimally small, hence P(X=5) = 0. It is meaningful to talk about P(X ≤ 5).

8.5 Probability Density Function… A function f(x) is called a probability density function (over the range a ≤ x ≤ b if it meets the following requirements: 1)f(x) ≥ 0 for all x between a and b, and 2)The total area under the curve between a and b is 1.0 f(x) xba area=1

8.6

8.7 Uniform Distribution… Consider the uniform probability distribution (sometimes called the rectangular probability distribution). It is described by the function: f(x) xba area = width x height = (b – a) x = 1

8.8 Example The amount of petrol sold daily at a service station is uniformly distributed with a minimum of 2,000 litres and a maximum of 5,000 litres. What is the probability that the service station will sell at least 4,000 litres? Algebraically: what is P(X ≥ 4,000) ? P(X ≥ 4,000) = (5,000 – 4,000) x (1/3000) =.3333 f(x) x5,0002,000

Bin width 25

Bin width 5

Bin width 1

Conditions for use of the Normal Distribution The data must be continuous (or we can use a continuity correction to approximate the Normal) The parameters must be established from a large number of trials

8.14 The Normal Distribution… The normal distribution is the most important of all probability distributions. The probability density function of a normal random variable is given by: It looks like this: Bell shaped, Symmetrical around the mean …

8.15 The Normal Distribution… Important things to note: The normal distribution is fully defined by two parameters: its standard deviation and mean Unlike the range of the uniform distribution (a ≤ x ≤ b) Normal distributions range from minus infinity to plus infinity The normal distribution is bell shaped and symmetrical about the mean

8.16 Standard Normal Distribution… A normal distribution whose mean is zero and standard deviation is one is called the standard normal distribution. Any normal distribution can be converted to a standard normal distribution with simple algebra. This makes calculations much easier

8.17 Normal Distribution… Increasing the mean shifts the curve to the right…

8.18 Normal Distribution… Increasing the standard deviation “ flattens ” the curve…

8.19 Calculating Normal Probabilities… Example: The time required to build a computer is normally distributed with a mean of 50 minutes and a standard deviation of 10 minutes: What is the probability that a computer is assembled in a time between 45 and 60 minutes? Algebraically speaking, what is P(45 < X < 60) ? 0

8.20 Calculating Normal Probabilities… P(45 < X < 60) ? 0 …mean of 50 minutes and a standard deviation of 10 minutes…

Tripthi M. Mathew, MD, MPH Distinguishing Features The mean ± 1 standard deviation covers 66.7% of the area under the curve The mean ± 2 standard deviation covers 95% of the area under the curve The mean ± 3 standard deviation covers 99.7% of the area under the curve

Rule 68% of the data 95% of the data 99.7% of the data

Are my data “ normal ” ? Not all continuous random variables are normally distributed!! It is important to evaluate how well the data are approximated by a normal distribution

Are my data normally distributed? 1.Look at the histogram! Does it appear bell shaped? 2.Compute descriptive summary measures—are mean, median, and mode similar? 3.Do 2/3 of observations lie within 1 std dev of the mean? Do 95% of observations lie within 2 std dev of the mean?

June 5, 2008 Stat Lecture 7 - Normal Distribution 25 Law of Large Numbers Rest of course will be about using data statistics (x and s 2 ) to estimate parameters of random variables (  and  2 ) Law of Large Numbers: as the size of our data sample increases, the mean x of the observed data variable approaches the mean  of the population If our sample is large enough, we can be confident that our sample mean is a good estimate of the population mean!

Points of note: Total area = 1 Only have a probability from width – For an infinite number of z scores each point has a probability of 0 (for the single point) Typically negative values are not reported – Symmetrical, therefore area below negative value = Area above its positive value Always draw a sketch!