Normal and Other Continuous distributions

Continuous probability distributions vary by the shape under the curve.

Normal Distribution

The normal distribution is not only symmetrical, but bell-shaped, a shape that (loosely) suggests the profile of a bell. Being bell-shaped means that most values of the continuous vari- able will cluster around the mean.

Uniform Distribution

It is also known as rectangular distribution, contains the values that are equally distributed in the range between the minimum value and the maximum value. In a uniform distribution every value has a equal chance of occurrence

Exponential Distribution

The exponential distribution contains values from zero to positive infinity and is right-skewed, making the mean greater than the median

Properties of Normal Distribution

• Symmetrical distribution. Its mean and median are therefore equal
• Bell-shaped. Values cluster around the mean
• Interquartile range is roughly 1.33 standard deviations. Therefore, the middle 50% of the values are contained within an interval that is approximately two-thirds of a standard deviation below and two-thirds of a standard deviation above the mean
• The distribution has an infinite range \(-\infty < X < \infty\)
• Six standard deviations approximate this range

Normal Probability Density Function

\(\large f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^-(\frac{1}{2})(\frac{x-\mu}{\sigma})^2\)

where,
e = mathematical constant (2.71828)
\(\pi\) = mathematical constant (3.14159)
\(\mu\) = mean
\(\sigma\) = standard deviation
X = Any value of the continuous variable, \(-\infty < X < \infty\)

Computing Normal Probabilities

To compute Normal probabilites, you need to first convert X to a standardized normal variable (Z), using Transformation formula

Z Transformation formula

The Z value is equal to difference between X and the mean,\(\mu\), divided by the standard deviation(\(\sigma\))

\(\text{Z =} \large \frac{x-\mu}{\sigma}\)

Excercises

1. What is the probability that the load time of a web sales page will be more than 9 seconds if average load time is 7 seconds with a standard deviation of 2?

1-pnorm(9,7,2)

## [1] 0.1586553

2. What is the probability that the load time for the web sales page will be less than 7 seconds or more than 9 seconds?

pnorm(7,7,2) + (1-pnorm(9,7,2))

## [1] 0.6586553

3. How much time (in seconds) will elapse before the fastest 10% of the web sales pages load time occur?

qnorm(0.10,7,2)

## [1] 4.436897

Thus, 10% of the load times are 4.44 seconds or less

Evaluating Normality

• Construct charts and observe their appearance. For small or moderate sized data sets, create a stem-and-leaf display or a boxplot. For large data sets, in addition, plot a histogram or polygon.

• Compute descriptive statistics and compare these statistics with the theoretical proper- ties of the normal distribution. Compare the mean and median. Is the interquartile range approximately 1.33 times the standard deviation? Is the range approximately 6 times the standard deviation?

• Evaluate how the values are distributed. Determine whether approximately two-thirds of the values lie between the mean and \(\pm\) 1 standard deviation. Determine whether approximately four-fifths of the values lie between the mean and \(\pm\) 1.28 standard deviations. Determine whether approximately 19 out of every 20 values lie between the mean and \(\pm\) 2 standard deviations.

The Uniform Distribution

In the uniform distribution, the values are evenly distributed in the range between the smallest value, a, and the largest value, b.

Because of its shape, the uniform distribution is sometimes called the rectangular distribution

Uniform Probability Density Function

\(\large f(x) = \Large \frac{1}{b-a}, \normalsize \text{if a} \le \text{X} \le b \land \text{0 elsewhere}\)

where,
a = minimum value of X
b = maximum value of X