Discrete Probability Distributions : Discreet Probability Description, Binomial Distribution

Discrete vs Continuous Variable Identifier

Random Variable Definition: 

A random variable is a mathematical concept used in probability theory. In simple terms, it is a variable whose value is determined by chance or randomness. 

Let me give you an example to help you understand better. Imagine that we have a bag containing five different colors of balls: red, blue, green, yellow, and purple. We want to choose one ball at random from the bag without looking. 

In this scenario, the color of the ball we choose is a random variable. We don't know ahead of time which color we will get, and the outcome is determined by chance. The random variable can take on different values, which in this case are the different colors of the balls. 

Random variables are used in many different fields, including science, economics, and engineering, to model and analyze various types of random events.

Discreet vs Continuous Random Variables 

A discrete random variable is a variable that can take on a countable number of distinct values. For example, if you flip a coin three times, the number of heads you get can only be 0, 1, 2, or 3. This is a discrete random variable because the possible values are distinct, separate numbers.

On the other hand, a continuous random variable is a variable that can take on an infinite number of values within a range. For example, the height of people in a population is a continuous random variable because it can take on any value between the minimum and maximum height in the population. This means that there are an infinite number of possible values that the height variable can take on, and these values are not distinct or separate like in the case of a discrete variable.

In summary, the main difference between a discrete and continuous random variable is that a discrete variable can only take on a countable number of distinct values, while a continuous variable can take on an infinite number of values within a range.

Tips to Help Distinguish between Discreet vs Continuous Random Variables 

1. Countable vs Uncountable Values: A discrete variable is one where the possible values are separate and countable. For example, the number of siblings you have, the number of pets you own, or the number of pencils in your pencil case. A continuous variable is one where the possible values are not separate, but instead fall along a range of values. For example, your height, your weight, or the temperature outside.

2. Nature of the variable: Think about the nature of what you're measuring. If you can only get specific values, like whole numbers, then it's likely a discrete variable. If you can get any value along a range, then it's likely a continuous variable.

3. Probability distributions: A probability distribution is a way of showing the chances of getting different values of a variable. For a discrete variable, this is shown using a probability mass function, which is like a chart with bars that show the chances of getting each value. For a continuous variable, this is shown using a probability density function, which is like a curve that shows the chances of getting any value within a range.

4. Units of measurement: The units of measurement can also give you a clue. If the variable is measured in whole numbers, like 1, 2, 3, etc., then it's likely a discrete variable. If the variable is measured in decimals or fractions, then it's likely a continuous variable.

Examples of Discreet vs Continuous Random Variables 

Discrete Random Variables:

1. Number of siblings: This is a discrete random variable because it can only take on integer values (0, 1, 2, 3, etc.).

2. Number of children in a family: This is a discrete random variable because it can only take on integer values (0, 1, 2, 3, etc.).

3. Number of goals scored in a soccer match: This is a discrete random variable because it can only take on integer values (0, 1, 2, 3, etc.).

4. Number of heads obtained in a coin toss: This is a discrete random variable because it can only take on integer values (0 or 1).

5. Number of defects in a batch of products: This is a discrete random variable because it can only take on integer values (0, 1, 2, 3, etc.).

Continuous Random Variables:

1. Temperature of a room: This is a continuous random variable because it can take on any value within a certain range (e.g., between 20 and 25 degrees Celsius).

2. Weight of a person: This is a continuous random variable because it can take on any value within a certain range (e.g., between 50 and 100 kg).

3. Height of a person: This is a continuous random variable because it can take on any value within a certain range (e.g., between 150 and 200 cm).

4. Time taken to complete a task: This is a continuous random variable because it can take on any value within a certain range (e.g., between 0 and 60 minutes).

5. Amount of rainfall in a certain period: This is a continuous random variable because it can take on any value within a certain range (e.g., between 0 and 100 mm).

Mean, Variance, Standard Deviation of a Discrete Probability Distribution

Mean, Variance and Standard Deviation of a Discrete Distribution Formulas and Example

Mean:

The mean of a discrete probability distribution is a way to find the "average" value of the data. To find the mean, you need to multiply each possible value by its probability, and then add up all of the products. For example, if you have a distribution of possible test scores (50, 60, 70, 80), and their respective probabilities are (0.2, 0.3, 0.4, 0.1), then the mean would be calculated as follows:

Mean = (50 x 0.2) + (60 x 0.3) + (70 x 0.4) + (80 x 0.1) = 64

Formula:

μ = Σ (xi * pi)

Where:

μ = mean

Σ = sum of all possible values

xi = each possible value

pi = probability of each value

Variance:

Variance is a measure of how spread out the data is from the mean. To calculate the variance of a discrete probability distribution, you need to first find the mean (using the method described above). Then, you need to subtract the mean from each possible value, square the difference, and multiply it by the probability. Add up all of these products, and you'll have the variance. For example, using the same test score distribution as before, we can find the variance as follows:

Mean = 64

Variance = [(50-64)^2 x 0.2] + [(60-64)^2 x 0.3] + [(70-64)^2 x 0.4] + [(80-64)^2 x 0.1] = 84

Formula:

σ^2 = Σ [(xi - μ)^2 * pi]

Where:

σ^2 = variance

Σ = sum of all possible values

xi = each possible value

μ = mean

pi = probability of each value

Standard Deviation:

The standard deviation is another measure of how spread out the data is from the mean, but it is in the same units as the data. To find the standard deviation of a discrete probability distribution, you simply take the square root of the variance. Using the same example, we can find the standard deviation as follows:

Standard Deviation = √(84) = 9.17

Formula:

σ = √(σ^2)

Where:

σ = standard deviation

σ^2 = variance

Mean, Variance, Standard Deviation of a Binomial Distribution

 Conditions that need to be met in order to use the binomial distribution:

1. There are a fixed number of trials (n).

2. Each trial has only two possible outcomes: success or failure.

3. The probability of success (p) is constant for each trial.

4. The trials are independent.

Formulas for the mean, variance, and standard deviation of a binomial distribution:

Mean (μ) = n * p

where n is the number of trials and p is the probability of success in each trial.

Example: Suppose you flip a fair coin 10 times. The number of successful flips (heads) is a binomial random variable with n = 10 and p = 0.5. The mean would be μ = 10 * 0.5 = 5.

 Variance (σ^2) = n * p * (1 - p)

where n is the number of trials and p is the probability of success in each trial.

Example: Continuing from the previous example, the variance would be σ^2 = 10 * 0.5 * (1 - 0.5) = 2.5.

Standard deviation (σ) = √(n * p * (1 - p))

where n is the number of trials and p is the probability of success in each trial.

Example: Using the same values as the previous examples, the standard deviation would be σ = √(10 * 0.5 * (1 - 0.5)) = √2.5 ≈ 1.58.

Binomial Distribution Probability Calculator

Binomial Distribution  Formula

Sure. The binomial probability distribution is a discrete probability distribution that describes the probability of getting a certain number of successes in a sequence of n independent experiments, each with a constant probability of success p. The probability of getting k successes is given by the formula:

P(X=k) = nCk * p^k * (1-p)^(n-k)

where:

• n is the number of experiments

• k is the number of successes

• p is the probability of success

• nCk is the binomial coefficient, which is the number of ways to choose k successes from n experiments

The binomial probability distribution can be used to model a variety of situations, such as:

• The number of heads in n coin flips

• The number of people who pass a test

• The number of defective items in a batch of products

The binomial probability distribution is a powerful tool for analyzing data and making predictions. It is used in a wide variety of fields, including statistics, engineering, and business.

Binomial Distribution Example

For a binomial distribution with n = 8 and π = 0.31, determine the probabilities of the given events. Round your answers to 4 decimal places.

a. x = 4.

b. x ≤ 4.

c. x ≥ 5.