Understanding the Cumulative Distribution Function (CDF) of Tossing Coins

In probability theory, the Cumulative Distribution Function (CDF) is a powerful tool used to describe the probability that a random variable X is less than or equal to a specific value. In this article, we will explore the concept of CDF using the example of tossing three coins. Our aim is to find the CDF and get a deeper understanding of the underlying probabilities.

The Scenario: Tossing Three Coins

When tossing three coins, the possible outcomes are:

Three heads (HHH) Two heads and one tail (HHT, HTH, THH) One head and two tails (HTT, THT, TTH) Three tails (TTT)

The sample space is denoted as: S {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

Calculating Probabilities

The random variable X represents the number of tails obtained. We need to calculate the probability of X taking each of these values (0, 1, 2, and 3).

X P(X0) P(X1) P(X2) P(X3) 0 (no tails) 1/8 (HHH) 1 (one tail) 3/8 (HHT, HTH, THH) 2 (two tails) 3/8 (HTT, THT, TTH) 3 (three tails) 1/8 (TTT)

Defining the CDF

The cumulative distribution function (CDF) Fx is defined as:

P(X ≤ x)
The CDF gives the probability that the random variable X is less than or equal to a specific value x.

Finding F2

Our task is to find the value of F2, which represents the probability of getting 2 or fewer tails:

F2 P(X 0) P(X 1) P(X 2)
Substituting the probabilities:

F2 1/8 3/8 3/8 7/8
Thus, the correct answer is b. 7/8.

Interpreting the Results

In a purely mathematical context, the result is precise and accurate. However, in the real world or when interpreted through different lenses, the answer might vary. For example:

If you are a mathematician: The answer is exactly 7/8. If you are an engineer: The answer is approximately 3/8. If you are a Postmodernist Lit Crit: The answer is influenced by the perspective of Jacques Derrida, questioning the binary logic and seeking a more nuanced understanding.

Building the CDF

x Fx 0 1/8 1 1/8 3/8 4/8 1/2 2 1/8 3/8 3/8 7/8 3 1/8 3/8 3/8 1/8 1

The CDF shows how the probability accumulates as x increases. The values we calculated are:

F0 1/8 F1 4/8 1/2 F2 7/8 F3 1

Conclusion

The cumulative distribution function (CDF) is a critical concept in probability and statistics. By understanding how to calculate and interpret it, we gain a deeper insight into the behavior of random variables. This example with tossing three coins demonstrates the practical application of CDF and highlights the different interpretations of results in various academic fields.