Problem 49 - An Introduction to Mathematical Statistics and Its Applications

Question

A fair die is rolled four times. Let the random variable \(X\) denote the number of 6 's that appear. Find and graph the cdf for \(X\).

Step-by-Step Solution

Answer

The cumulative distribution function \(F(x)\) can be computed for each \(x=0, 1, 2, 3, 4\) using the formula \(F(x) = \sum_{k=0}^{x} P(X=k)\), where \(P(X=k) = C(n, k) \cdot {p^k} \cdot {(1-p)^{n-k}}\) given \(n=4\) and \(p=1/6\). These calculations can then be plotted on a graph to form a step function representing the CDF of \(X\).

Step 1 - Understanding binomial distribution

A binomial distribution can be used to solve this problem because rolling a die is an independent event, and each roll has two possible outcomes: rolling a 6 (which we can deem as 'success') with probability \(p=1/6\), or not rolling a 6 (which we can deem as 'failure') with probability \(q=1-p=5/6\). Here, \(X\) follows a Binomial distribution with parameters \(n=4\) and \(p=1/6\).

Step 2 - Writing formula for CDF

The cumulative distribution function \(F(x)\) of a random variable \(X\) is the probability that \(X\) will take a value less than or equal to \(x\), formally given by \(F(x)=P(X \leq x)\). For the binomial distribution, the CDF is given by the sum of the probabilities up to \(x\), \[F(x) = \sum_{k=0}^{x} P(X=k)\], where \(P(X=k)\) is the probability mass function of \(X\) and is given by \[P(X=k) = C(n, k) \cdot {p^k} \cdot {(1-p)^{n-k}}\], where \(C(n, k)\) is the binomial coefficient representing the number of ways to choose \(k\) successes out of \(n\) trials.

Step 3 - Compute CDF for all values of \(X\)

Since \(X\) can take on any value between \(0\) and \(n=4\), compute \(F(x)\) for \(x=0, 1, 2, 3, 4\) using the above formulas. Each calculation requires substituting the appropriate values for \(n, p, k\) into the formula for \(P(X=k)\), adding up these probabilities for each \(k\) from \(0\) to \(x\), and then computing the sum.

Step 4 - Graph CDF

Plot the points \((x, F(x))\) for \(x=0, 1, 2, 3,4\) on a graph. Connect the points to form a step function which increases on the intervals \(x \in [0,1), [1,2), [2,3), [3,4], [4,5]\). Note that since the probabilities are cumulative, the function is non-decreasing.