Continuous Probability Distribution: The Uniform Distribution, The Normal Distribution, The Standard Normal Distribution

The Uniform Probability Distribution Probability Description and Formulas

A continuous uniform probability distribution is a type of probability distribution where all outcomes are equally likely to occur. It is defined by two parameters, a minimum value and a maximum value. In other words, if you imagine a graph with the x-axis representing all possible outcomes and the y-axis representing the probability of each outcome, the graph would be a flat line between the minimum and maximum values.

Mean:

Mean (Expected Value) = (a+b)/2

Probability:

The probability density function (PDF) of a continuous uniform distribution on the interval [a, b] is:

P(x) = 1 / (b - a)   for a <= x <= b

P(x) = 0             for x < a or x > b

Variance:

The variance of a continuous uniform distribution on the interval [a, b] is given by:

Var(X) = [(b - a)^2] / 12

Standard Deviation:

The standard deviation of a continuous uniform distribution on the interval [a, b] is:

SD(X) = sqrt{[(b - a)^2] / 12}

where sqrt denotes the square root function.

Finding the Mean, Variance and Standard Deviation of a Uniform Probability Distribution Example

If customers encounter technical issues with their gas supply service, they can contact a technical support number that begins with 600. The time taken by the technician to resolve the problem falls within a range of 30 seconds to 120 minutes. The distribution of the problem resolution time is uniformly distributed.

a. What are the values for a and b in minutes?  

b. What is the mean time to resolve the problem? 

c. What is the variance of the time?  

d. What is the standard deviation of the time?

Finding  Values Associated to Probabilities - The Uniform Probability Distribution Guide and Example

Guide:

1. Figure out the range of values for the distribution: Find the lowest and highest possible values.

2. Calculate the total range of the distribution: Subtract the lowest value from the highest value.

3. Understand that in a uniform distribution, all values within the range are equally likely.

4. To find the probability associated with a specific value: Divide 1 by the total range.

5. To find the cumulative probability (probability up to a certain value): Divide the difference between the value and the lowest value by the total range.

6. To find the value associated with a specific probability: Multiply the total range by the desired probability, and add the result to the lowest value.

Remember, in a uniform distribution, all values within the range have the same chance of occurring.

Example

If customers encounter technical issues with their gas supply service, they can contact a technical support number. The time taken by the technician to resolve the problem falls within a range of 30 seconds to 120 minutes. The distribution of the problem resolution time is uniformly distributed.

a)What is the time by which the 10% of callers have their problem resolved in the shortest time?

b)What are the start and end times by which the middle 50% of calls are resolved?

Finding Probabilities Associated with Values - The Uniform Probability Distribution - Guide and Example

Guide

A uniform distribution is a probability distribution where all outcomes are equally likely. In this case, the probability of any specific value occurring within a given range is the same for all values within that range.

To find the probability associated with a value on a uniform probability distribution, you need to know the range of possible values and the interval width.

Here's a step-by-step guide to help you calculate probabilities:

Step 1: Determine the range of possible values: Define the minimum and maximum values for your uniform distribution. Let's denote them as "a" and "b," respectively.

Step 2: Calculate the interval width: Subtract the minimum value "a" from the maximum value "b" to determine the range of the distribution. Let's call this value "R."

   R = b - a

Step 3: Calculate the probability density function (PDF): In a uniform distribution, the probability density function is constant within the defined range. To calculate it, divide 1 by the range R.

   PDF = 1 / R

Step 4: Determine the probability associated with a specific value: To find the probability of a particular value "x" occurring within the range, subtract the minimum value "a" from "x" and divide the result by the range R.

   P(x) = (x - a) / R

Step 5: Verify the probability: Ensure that the probability obtained in Step 4 is between 0 and 1, inclusive. If it falls outside this range, double-check your calculations.

By following these steps, you can find the probability associated with a specific value on a uniform probability distribution. Remember that this method assumes a continuous uniform distribution. If you are working with a discrete uniform distribution, the calculations might differ slightly.

Example:

If customers encounter technical issues with their gas supply service, they can contact a technical support number. The time taken by the technician to resolve the problem falls within a range of 30 seconds to 120 minutes. The distribution of the problem resolution time is uniformly distributed.

a. What is the probability that a random caller, will have their problem resolved within 30 minutes?  

b. What is the probability that a random caller, will have their problem resolved within an hour?  

c. What is the probability that a random caller, will have their problem solved within an hour, but longer than 30 minutes?

The Normal Probability Distribution Description

Normal Probability Distribution and the Standard Normal Probability Description

The normal probability distribution, also known as the Gaussian distribution or bell curve, is a type of continuous probability distribution that is commonly used to model real-world phenomena such as heights, weights, test scores, and many other measurements. It has a characteristic bell-shaped curve that is symmetric around the mean or average value. The probability density function of the normal distribution is defined by two parameters: the mean (μ) and the standard deviation (σ). 

The standard normal probability distribution is a specific form of the normal distribution where the mean is 0 and the standard deviation is 1. The probability density function of the standard normal distribution is denoted by the Greek letter phi (Φ) and is defined by the following formula:

Φ(z) = (1/√(2π)) * e^(-z^2/2)

where z is a standard normal random variable, e is the mathematical constant 2.71828..., and π is the mathematical constant pi (3.14159...). The standard normal distribution is important because it allows us to convert any normal distribution into a standard normal distribution, which makes it easier to compare and analyze data.

The normal distribution is a common pattern that many things in nature and society tend to follow. It is often referred to as the "bell curve" because when graphed, it creates a symmetrical shape that looks like a bell.

In a normal distribution, the data is evenly spread around a central value, which is the mean. The majority of the data points cluster around the mean, and as you move away from the mean, the number of data points gradually decreases. This pattern creates the bell-shaped curve.

The normal distribution has some important characteristics:

1. It is symmetric: The curve is perfectly balanced on both sides of the mean.

2. The mean, median, and mode are all the same and located at the center of the distribution.

3. It is continuous: The distribution represents a range of possible values, not specific individual values.

4. It is defined by two parameters: the mean (μ) and the standard deviation (σ). The mean determines the center of the distribution, while the standard deviation determines the spread or variability of the data points around the mean.

Many real-world phenomena can be approximated by a normal distribution, such as heights and weights of individuals, IQ scores, and measurement errors. The normal distribution is widely used in statistics and is the basis for many statistical tests and models.

The Standard Normal Probability - How to Find Z-Scores Guide and Example

The Z-score is a statistical measure that tells us how many standard deviations a particular data point is away from the mean of a distribution. It helps us understand the relative position of a data point within a dataset.

To convert a normal distribution into a standard normal distribution, we use a process called standardization. This involves subtracting the mean from each data point and dividing by the standard deviation. This transforms the original data into a standard normal distribution with a mean of 0 and a standard deviation of 1. The resulting values are called z-scores, and they indicate how many standard deviations a given data point is from the mean. 

The formula to calculate the Z-score for a data point is:

Z = (X - μ) / σ

Where:

- Z is the Z-score

- X is the data point you want to calculate the Z-score for

- μ is the mean of the distribution

- σ is the standard deviation of the distribution

By subtracting the mean from the data point and dividing it by the standard deviation, we get a standardized value that allows us to compare the data point to the rest of the distribution. The Z-score tells us how many standard deviations the data point is above or below the mean. Positive Z-scores indicate that the data point is above the mean, while negative Z-scores indicate that the data point is below the mean.

The Standard Normal Probability - Example

A normal population has a mean of 18 and a standard deviation of 3. 

a. Compute the z value associated with 21.  

b. Compute the z value associated with 15.

c. Compute the z value associated with 24.

Finding  Value Associated with Probabilities using the Standard Normal Distribution Guide and Example

Guide: 

1. Understand the normal distribution: The normal distribution, also known as the bell curve, is a common probability distribution that is symmetric and bell-shaped. It is characterized by its mean (μ) and standard deviation (σ).

2. Determine the mean and standard deviation: Identify the given mean and standard deviation values for the normal distribution you are working with.

3. Use the Z-score formula: To find the value associated with a specific probability, you need to convert the probability into a standardized value called the Z-score.

   - Calculate the Z-score: Subtract the mean (μ) from the value you're interested in and divide the result by the standard deviation (σ): Z = (x - μ) / σ.

4. Use a Z-score table or calculator: Look up the Z-score in a Z-score table or use a Z-score calculator to find the corresponding percentile or probability associated with that Z-score.

By following these steps, you can find a value associated with a probability using the normal distribution. Remember to determine the mean and standard deviation, calculate the Z-score, use a Z-score table or calculator to find the corresponding percentile or probability, and reverse the process to find the value associated with a specific probability.

Example: 

A normal population has a mean of 18 and a standard deviation of 3. 

a. The bottom 25%  of the population values are less than what value? 

b. The top 25% of the population values are greater than what value? 

c. The middle 50% of the population values occur between which 2 values? 

Finding Probability Associated with Values - The Normal Probability Distribution - Guide and Example

Guide: 

1. Understand the normal distribution: The normal distribution, also known as the bell curve, is a common probability distribution that is symmetric and bell-shaped. It is characterized by its mean (μ) and standard deviation (σ).

2. Determine the mean and standard deviation: Identify the given mean and standard deviation values for the normal distribution you are working with.

3. Use the Z-score formula: To find the probability associated with a specified value, you need to convert the value into a standardized value called the Z-score.

   - Calculate the Z-score: Subtract the mean (μ) from the specified value and divide the result by the standard deviation (σ): Z = (x - μ) / σ.

4. Use a Z-score table or calculator: Look up the Z-score in a Z-score table or use a Z-score calculator to find the corresponding percentile or probability associated with that Z-score.

   - If using a table, locate the Z-score in the table and find the corresponding probability.

   - If using a calculator, input the Z-score and obtain the probability directly.

5. Understand the probability interpretation: The probability you find represents the area under the normal curve to the left (or right, depending on the context) of the specified value.

By following these steps, you can find the probability associated with a specified value using the normal probability distribution. Remember to determine the mean and standard deviation, calculate the Z-score, use a Z-score table or calculator to find the corresponding percentile or probability, and interpret the probability as the area under the normal curve.

Example:

A normal population has a mean of 18 and a standard deviation of 3. 

a. What proportion of the population is between 20 and 22? (Round z-score computation to 2 decimal places and your final answer to 4 decimal places.) 

b. What proportion of the population is less than 17? (Round z-score computation to 2 decimal places and your final answer to 4 decimal places.) 

c. What proportion of the population is greater than 24? (Round z-score computation to 2 decimal places and your final answer to 4 decimal places.)