Plot a graph of the binomial frequency function \(f(x)\) for the case \(n=6, p=\frac{1}{6}, q=\frac{5}{6}\), representing the probability of, say, \(x\) aces in 6 throws of a die. Also draw graphs of \(n f(x)\) as a function of \(x / n\), and of \(F(x)\). What is the probability of at least 2 aces out of 6 tosses of a die?

The Binomial Probability Function describes the probability of obtaining a specific number of successes in a fixed number of trials of a binary experiment. The function is represented as follows:
\( f(x) = \binom{n}{x} p^x q^{n-x} \) Here, \( n \) is the number of trials, \( x \) is the number of successes, \( p \) is the probability of success in a single trial, and \( q \) (equal to \( 1 - p \)) is the probability of failure.
For instance, in the exercise where we throw a die 6 times (=6\therapy\text{\frac{1}{6}\text{\frac{5}{6})}).
To find the probability of, for example, getting exactly 2 aces (success), we substitute the values into the functional form:
<\( binom{6}{2}\frac{1}{6}^2\frac{5}{6}^4\)
This step-by-step calculation helps determine the likelihood of each possible number of aces from 0 to 6.

The Cumulative Distribution Function (CDF) for a binomial distribution calculates the probability that a random variable \(X\) is less than or equal to a certain value. It is represented by:

F(x) = P(X ≤ x)
To find the probability of having up to a particular number of successes, sum up the probabilities for all outcomes from 0 to that number.
For the example exercise, this means summing probabilities calculated by the probability function from 0 to x.
For instance, finding \(F(2)\) means calculating and summing \(f(0)\), \(f(1)\), and \(f(2)\) using the binomial probability function.
The cumulative probabilities build upon each other and give a comprehensive view of the distribution up to each point.

The Binomial Coefficient is a key component in the binomial probability function, represented by the symbol \( \binom{n}{x} \). It is calculated as:

\( \binom{n}{x} = \frac{n!}{x!(n-x)!} \)
This formula determines the number of ways to choose \( x \) successes out of \( n \) trials, taking into account the order of selection.
For example, when calculating probabilities for our die-throwing exercise, computing \( \binom{6}{2} \) would involve:

\( \binom{6}{2} = \frac{6!}{2!(6-2)!} \)

\( = \frac{6 \times 5 \times 4!}{2 \times 1 \times 4!} \)

= 15
These coefficients are essential in determining the individual probabilities used in the binomial distribution and the cumulative distribution function calculations.

Visualizing the Binomial Distribution through graph plotting helps understand the data and distribution properties.
In the described exercise, you effectively plot three different graphs:

• The Binomial Probability Function (f(x)) against x:
  Displays the probability for each possible number of aces (successes).
• nf(x) against x/n:
  Normalizes the distribution, making it easier to compare different binomial distributions.
• The Cumulative Distribution Function (F(x)):
  Illustrates the accumulated probability up to each value of x.

Each of these graphs offers unique insights and helps compare how probabilities accumulate and distribute across different values of x. This comprehensive visual representation aids in better understanding binomial distribution behavior and data interpretation.