Advanced Combinatorial Discrete Models

00:00 7 marks

Consider the following experiment: A fair coin is tossed n times, and X denotes the number of heads observed.

(a) Establish a discrete model for X using combinatorial reasoning.

(b) Using the model from part (a), derive the mean of the binomial distribution and show that it is given by \(\mu = np\).

Part a

5 minutes 3 marks

Question

Establish a discrete model for the number of heads \(X\) observed when a fair coin is tossed n times. In your answer, define the probability mass function (PMF) for \(X\) and explain the role of the binomial coefficient in calculating \(P(X=k)\).

Hint Hint

Hint

Recall the binomial formula \(P(X=k)=\binom{n}{k}(0.5)^k(0.5)^{n-k}\) for a fair coin toss.

Answer Answer
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Answer Answer

Complete Answer

X is modeled as a binomial random variable with n trials and success probability \(p=0.5\). Its PMF is given by \(P(X=k)=\binom{n}{k}(0.5)^k(0.5)^{n-k}\), for \(k=0,1,\ldots,n\).

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Part b

5 minutes 4 marks

Question

Using the model from part (a), derive the expected value \(E[X]\) of the binomial random variable \(X\) and show that it equals \(np\). Outline each step of your derivation with combinatorial reasoning.

Hint Hint

Hint

Start with the definition \(E[X]=\sum_{k=0}^{n} kP(X=k)\) and use properties of binomial coefficients.

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Complete Answer

By applying the definition of expectation and using the identity \(k\binom{n}{k}=n\binom{n-1}{k-1}\), we deduce that \(E[X]=np\).

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