# Chapter 8 : Continuous Probability

### Topics covered in this snack-sized chapter:

A random variable is called continuous if it can assume all possible values in the possible range of the random variable.
Probability Distribution of random variable is known as continuous probability distribution.
##### Examples:

- Time required to complete a task.

- Temperature of a solution.

Continuous Probability Distribution includes the following:

The Probability of the Random Variable assuming a value within some given interval from x_{1
} to x_{2
} is defined to be the area under the graph of the probability density function between x_{1
} and x_{2
}.
Probability is the area under the curve.

A random variable is uniformly distributed whenever the probability is proportional to the interval’s length.
The uniform probability density function is given below:

Where,
- f(x) = Value of the density function at any x value.

The uniform distribution is a probability distribution that has equal probabilities for all possible outcomes of the random variable.
Expected Value (mean) of x.

Variance of x.

The Normal Probability Distribution is the most important distribution for describing a continuous random variable.
It is widely used in statistical inference:

###### Properties of Normal Distribution:

Bell Shaped.
Symmetrical.
Mean = Median = Mode.
Location is determined by the mean, .
Spread is determined by the standard deviation, .
The random variable has an infinite theoretical range: .
- Changing shifts the distribution left or right.

- Changing increases or decreases the spread.

###### Characteristics of the Normal Distribution:

The entire family of normal probability distributions is defined by its mean m and its standard deviations.
The distribution is symmetric if its Skewness measure is zero.

The highest point on the normal curve is at the mean, which is also the median and mode.
The mean can be any numerical value: negative, zero, or positive.
The standard deviation determines the width of the curve: larger values result in wider, flatter curves.
Probabilities for the normal random variable are given by areas under the curve.
- The total area under the curve is 1 (.5 to the left of the mean and .5 to the right)

For most data sets:
- 68% of the values of a normal random variable are within 1 Standard Deviation (SD) of its mean.

- 95% of the values of a normal random variable are within 2 SD of its mean.

- 99.7% of the values of a normal random variable are within (±) 3 SD of its mean.

Where,
- Value of the density function at any value.

- The mathematical constant approximated by 2.71828.

- The mathematical constant approximated by 3.14159.

- The population mean.

- The population standard deviation.

- Any value of the continuous variable.

A random variable having a normal distribution with a mean of 0 and a standard deviation of 1 is said to have a Standard Normal Probability Distribution.
Any normal distribution (with any mean and standard deviation combination) can be transformed into the standardized normal distribution.
- Also known as the distribution.

- Values above the mean have positive Z-values.

- Values below the mean have negative Z-values.

We can think ofz as a measure of the number of standard deviations x is from .
We use the above equation to convert normal distribution into standard normal distribution.

Where,
- Any value of the continuous variable.

- Value of the density function at any value.

- The mathematical constant approximated by 2.71828.

- The mathematical constant approximated by 3.14159.

It is used to model the length of time between two occurrences of an event (the time between arrivals).

##### Example:

Time between transactions at an ATM Machine.