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Normal Distribution

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Normal Distribution

Introduction

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.

- 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).

- A function f(x) is called a probability density function (over the range a ≤ x ≤ b if it meets the following requirements:
- f(x) ≥ 0 for all x between a and b, and
- The total area under the curve between a and b is 1.0

f(x)

area=1

a

b

x

- Consider the uniform probability distribution (sometimes called the rectangular probability distribution).
- It is described by the function:

f(x)

a

b

x

area = width x height = (b – a) x = 1

- 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)

2,000

5,000

x

- 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

- 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 …

- Important things to note:

The normal distribution is fully defined by two parameters:

its standard deviation andmean

The normal distribution is bell shaped and

symmetrical about the mean

Unlike the range of the uniform distribution (a ≤ x ≤ b)

Normal distributions range from minus infinity to plus infinity

0

1

1

- 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.

- Increasing the mean shifts the curve to the right…

- Increasing the standard deviation “flattens” the curve…

- 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

…mean of 50 minutes and a

standard deviation of 10 minutes…

- P(45 < X < 60) ?

0

- 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

Tripthi M. Mathew, MD, MPH

68% of the data

95% of the data

99.7% of the data

- Not all continuous random variables are normally distributed!!
- It is important to evaluate how well the data are approximated by a normal distribution

- Look at the histogram! Does it appear bell shaped?
- Compute descriptive summary measures—are mean, median, and mode similar?
- Do 2/3 of observations lie within 1 stddev of the mean? Do 95% of observations lie within 2 stddev of the mean?

- Rest of course will be about using data statistics (x and s2) 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!

Stat 111 - Lecture 7 - Normal Distribution

- 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!