Marginal Probability

Marginal Probability is a fundamental concept in probability theory and statistics. It refers to the probability of the single event occurring irrespective of the outcomes of the other related events. The Marginal probabilities are essential for understanding joint distributions and are commonly used in various fields including economics, engineering and social sciences.

What is Marginal Probability?

Marginal Probability refers to the probability of a single event occurring, without consideration of any other events. It is derived from a joint probability distribution and represents the likelihood of an event happening in isolation.

For example, if you have a deck of cards, the marginal probability of drawing a red card (either hearts or diamonds) is calculated by considering only the total number of red cards (26 out of 52). Therefore, the marginal probability of drawing a red card is 26/52, which simplifies to 1/2 or 0.5.

In simpler terms, marginal probability answers the question: "What is the chance of this one event happening, regardless of anything else?"

Formula for Marginal Probability

Formula for marginal probability are different for different variables:

• Discrete Random Variables
• Continuous Random Variables

For Discrete Random Variables

Let X and Y be two discrete random variables with the joint probability mass function P(X = x, Y = y). The marginal probability mass function of X is obtained by summing over all possible values of Y:

P(X = x) = \sum_{y} P(X = x, Y = y)

Similarly, the marginal probability mass function of Y is:

P(Y = y) = \sum_{x} P(X = x, Y = y)

For Continuous Random Variables

For continuous random variables, the joint probability density function f_{X,Y}(x, y) is used. The marginal probability density function of the X is obtained by the integrating over all values of Y:

f_X(x) = \int_{-\infty}^{\infty} f_{X,Y}(x, y) \, dy

The marginal probability density function of Y is:

f_Y(y) = \int_{-\infty}^{\infty} f_{X,Y}(x, y) \, dx

How to Calculate Marginal Probability?

To derive marginal probability from the joint probability distribution follow these steps:

Step 1: Identify the Joint Probability Distribution

Obtain or calculate the joint probability distribution for the random variables in question. This could be a table or mathematical function.

Step 2: Sum or Integrate Over the Other Variables

For discrete random variables: The Sum the joint probabilities over all possible values of the other variables. For example, if P(X, Y) is the joint probability of the X and Y the marginal probability of the X is:

P(X) = \sum_{y} P(X, Y = y)

For continuous random variables: The Integrate the joint probability density function over the range of the other variables. For example, if f_{X,Y}(x, y) is the joint probability density function the marginal density function of the X is:

f_X(x) = \int_{-\infty}^{\infty} f_{X,Y}(x, y) \, dy

Step 3: Interpret the Marginal Probability

The resulting value represents the probability of the event related to the one variable occurring the irrespective of the values of the other variables.

Read More about How to Calculate Marginal Probability.

Marginal Probability vs. Conditional and Joint Probability

To understand marginal probability better it's essential to the differentiate it from the joint and conditional probability:

Joint Probability

The Joint probability refers to the probability of the two or more events occurring the simultaneously. For example, the probability of both the A and B occurring is denoted as the P(A \cap B) or P(A, B).

Example: If we have two dice the joint probability of the rolling a 3 on the first die and 4 on the second die is P(\text{Die 1} = 3 \text{ and } \text{Die 2} = 4).

Marginal Probability

The Marginal probability is the probability of the single event occurring without considering the other events. It is derived from the joint probability distribution by the summing or integrating out the other variables.

Example: From the joint distribution of the rolling two dice the marginal probability of the rolling a 3 on the first die is obtained by the summing the joint probabilities for the all possible outcomes of the second die.

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. It is denoted as the P(A \mid B) which represents the probability of the event A occurring given that event B has occurred.

Example: The probability of rolling the 3 on the first die given that the second die shows a 4 is P(\text{Die 1} = 3 \mid \text{Die 2} = 4).

Read More about Probability: Joint vs. Marginal vs. Conditional.

Applications of Marginal Probability

The Marginal probabilities are widely used in the various applications:

• Statistical Inference: The Marginal probabilities help in estimating the distribution of the individual variables from the joint distributions.
• Decision Making: In fields like finance and engineering, marginal probabilities assist in making the decisions based on the likelihood of the individual events.
• Data Analysis: They are crucial for the analyzing complex datasets where multiple variables are involved.

Conclusion

Marginal probability is a critical concept that provides the insights into the behavior of the individual random variables within a joint distribution. By focusing on the one variable at a time, it simplifies the analysis of the complex probabilistic models and aids in the various practical applications. Understanding and calculating marginal probabilities is essential for the anyone working in fields that involve statistical analysis and probability theory.

Read More,

• Probability in Maths
• Coin Probability
• Joint Probability
• Conditional Probability
• Probability Density Function(PDF)
• Marginal Distribution

Solved Examples

Example 1: Consider the joint probability mass function of the two discrete random variables X and Y:

\begin{array}{c|cc} P(X, Y) & Y = 1 & Y = 2 \\ \hline X = 1 & 0.2 & 0.3 \\ X = 2 & 0.1 & 0.4 \\ \end{array}

Find the marginal probability P(X = 1) and P(Y = 2).

Solution:

To find P(X = 1):

P(X = 1) = P(X = 1, Y = 1) + P(X = 1, Y = 2) = 0.2 + 0.3 = 0.5

To find P(Y = 2):

P(Y = 2) = P(X = 1, Y = 2) + P(X = 2, Y = 2) = 0.3 + 0.4 = 0.7

Example 2: Suppose X and Y are continuous random variables with the joint density function:

f_{X,Y}(x, y) = \frac{1}{8} e^{-\frac{x + y}{4}}

Find the marginal density functions f_X(x) and f_Y(y).

Solution:

To find f_X(x):

f_X(x) = \int_{0}^{\infty} \frac{1}{8} e^{-\frac{x + y}{4}} \, dy = \frac{1}{8} e^{-\frac{x}{4}} \int_{0}^{\infty} e^{-\frac{y}{4}} \, dy

Since

\int_{0}^{\infty} e^{-\frac{y}{4}} \, dy = 4,

we have

f_X(x) = \frac{1}{2} e^{-\frac{x}{4}}

To find f_Y(y):

f_Y(y) = \int_{0}^{\infty} \frac{1}{8} e^{-\frac{x + y}{4}} \, dx = \frac{1}{8} e^{-\frac{y}{4}} \int_{0}^{\infty} e^{-\frac{x}{4}} \, dx

Since

\int_{0}^{\infty} e^{-\frac{x}{4}} \, dx = 4,

we have

f_Y(y) = \frac{1}{2} e^{-\frac{y}{4}}

Example 3: The joint probability distribution of the X and Y is given by:

P(X = x, Y = y) = \frac{1}{12} \text{ for } x = 1, 2 \text{ and } y = 1, 2

Find P(X = 2) and P(Y = 1).

Solution:

To find P(X = 2):

P(X = 2) = P(X = 2, Y = 1) + P(X = 2, Y = 2) = \frac{1}{12} + \frac{1}{12} = \frac{2}{12} = \frac{1}{6}

To find P(Y = 1):

P(Y = 1) = P(X = 1, Y = 1) + P(X = 2, Y = 1) = \frac{1}{12} + \frac{1}{12} = \frac{2}{12} = \frac{1}{6}

Example 4: Let the joint probability mass function be:

\begin{array}{c|ccc} P(X, Y) & Y = 0 & Y = 1 & Y = 2 \\ \hline X = 0 & 0.1 & 0.2 & 0.1 \\ X = 1 & 0.2 & 0.3 & 0.1 \\ \end{array}

Find the marginal probabilities P(X = 1) and P(Y = 0).

Solution:

To find P(X = 1):

P(X = 1) = P(X = 1, Y = 0) + P(X = 1, Y = 1) + P(X = 1, Y = 2) = 0.2 + 0.3 + 0.1 = 0.6

To find P(Y = 0):

P(Y = 0) = P(X = 0, Y = 0) + P(X = 1, Y = 0) = 0.1 + 0.2 = 0.3

Example 5: Given the joint density function:

f_{X,Y}(x, y) = \frac{1}{4} e^{-\frac{x + 2y}{2}}

Find the marginal density functions f_X(x) and f_Y(y).

Solution:

To find f_X(x):

f_X(x) = \int_{0}^{\infty} \frac{1}{4} e^{-\frac{x + 2y}{2}} \, dy = \frac{1}{4} e^{-\frac{x}{2}} \int_{0}^{\infty} e^{-y} \, dy

Since

\int_{0}^{\infty} e^{-y} \, dy = 1,

we have

f_X(x) = \frac{1}{4} e^{-\frac{x}{2}} \cdot 1 = \frac{1}{4} e^{-\frac{x}{2}}

To find f_Y(y):

f_Y(y) = \int_{0}^{\infty} \frac{1}{4} e^{-\frac{x + 2y}{2}} \, dx = \frac{1}{4} e^{-y} \int_{0}^{\infty} e^{-\frac{x}{2}} \, dx

Since

\int_{0}^{\infty} e^{-\frac{x}{2}} \, dx = 2,

we have

f_Y(y) = \frac{1}{4} e^{-y} \cdot 2 = \frac{1}{2} e^{-y}

Practice Questions: Marginal Probability

Q1: Given a joint probability mass function P(X, Y) how do you compute P(X = x)?

Q2: How do you find the marginal probability P(Y = y) if X and Y are discrete random variables?

Q3: For a continuous joint density function f_{X, Y}(x, y) how is the marginal density f(x) computed?

Q4: If X and Y are independent random variables what is the relationship between their joint and marginal distributions?

Q5: How do you determine the marginal probability P(X = x) from the joint probability table?

Q6: How do you calculate P(X ≤ x) from the marginal density function f X(x)?

Q7: Given the joint probability mass function P(X, Y) how do you compute P(X > x)?

Q8: How do you compute the marginal distribution of the X from the given joint distribution involving the X and Y?

Q9: What steps are involved in finding the marginal density of the Y for the continuous random variables?