# Chapter 8 : Continuous Probability

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

#### Continuous Probability arrow_upward

• 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:
• Thickness of an item.
• Time required to complete a task.
• Temperature of a solution.
• Height, in inches.
• Continuous Probability Distribution includes the following:
• #### Probability of the Random Variable arrow_upward

• The Probability of the Random Variable assuming a value within some given interval from x1 to x2 is defined to be the area under the graph of the probability density function between x1 and x2 .
• Probability is the area under the curve.
•  #### Uniform Distribution arrow_upward

• 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.
• a = Minimum value of x.
• b = Maximum value of x.
• 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.
• #### Normal Distribution arrow_upward

• The Normal Probability Distribution is the most important distribution for describing a continuous random variable.
• It is widely used in statistical inference:
• Height of the people.
• Amount of rainfall.
• • ###### 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. #### Normal Distribution: Formula arrow_upward • 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.

#### Standardized Normal Distribution arrow_upward

• 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.
• Mean is 0.
• Standard Deviation is 1.
• Values above the mean have positive Z-values.
• Values below the mean have negative Z-values. • We can think of z 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.
• #### Standard Normal Distribution: Formula arrow_upward • 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.

#### Exponential Distribution arrow_upward

• 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.

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