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Chapter 2: Problem 18

Show that when \(r=2\) the multinomial reduces to the binomial.

Short Answer

Expert verified
When \(r=2\), the multinomial theorem becomes \(\begin{pmatrix}n\\ m_1, m_2 \end{pmatrix}=\frac{n!}{m_1!m_2!}\) with \(m_2=n-m_1\). Replacing \(m_2\) with \(n-m_1\) gives us \(\begin{pmatrix}n\\ m_1, n - m_1 \end{pmatrix}=\frac{n!}{m_1!(n-m_1)!}\), which is equivalent to the binomial theorem \(\binom{n}{m_1}=\frac{n!}{m_1!(n-m_1)!}\). Hence, when \(r=2\), the multinomial reduces to the binomial.

Step by step solution

01

Input \(r=2\) into the Multinomial Theorem

When we input \(r=2\) into the multinomial theorem, we have two types of events (\(m_1\) and \(m_2\)). The multinomial becomes: \[ \begin{pmatrix}n\\ m_1, m_2 \end{pmatrix}=\frac{n!}{m_1!m_2!}. \] Since we have only 2 types of events and the total number of events is \(n\), we can express the relationship between \(m_1\) and \(m_2\) as \[ m_2=n-m_1\]
02

Replace \(m_2\) with \(n - m_1\)

Now that we have expressed the relationship between \(m_1\) and \(m_2\), we will replace \(m_2\) with \(n - m_1\) in the multinomial: \[ \begin{pmatrix}n\\ m_1, n - m_1 \end{pmatrix}=\frac{n!}{m_1!(n-m_1)!}. \]
03

Show Equivalence to the Binomial Theorem

Now let's compare our reduced multinomial to the binomial theorem. We can rewrite the binomial theorem with \(k = m_1\): \[\binom{n}{m_1}=\frac{n!}{m_1!(n-m_1)!}.\] Thus, we have: \[ \begin{pmatrix}n\\ m_1, n - m_1 \end{pmatrix}=\frac{n!}{m_1!(n - m_1)!} = \binom{n}{m_1}. \] So we have shown that when \(r=2\), the multinomial reduces to the binomial.

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Most popular questions from this chapter

Let \(X_{1}, X_{2}, \ldots\) be a sequence of independent identically distributed continuous random variables. We say that a record occurs at time \(n\) if \(X_{n}>\max \left(X_{1}, \ldots, X_{n-1}\right)\) That is, \(X_{n}\) is a record if it is larger than each of \(X_{1}, \ldots, X_{n-1}\). Show (a) \(P(\) a record occurs at time \(n\\}=1 / n ;\) (b) \(E[\) number of records by time \(n]=\sum_{i=1}^{n} 1 / i ;\) (c) \(\operatorname{Var}(\) number of records by time \(n)=\sum_{i=1}^{n}(i-1) / i^{2}\); (d) Let \(N=\min (n: n>1\) and a record occurs at time \(n] .\) Show \(E[N]=\infty\). Hint: For (ii) and (iii) represent the number of records as the sum of indicator (that is, Bernoulli) random variables.

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Use Chebyshev's inequality to prove the weak law of large numbers. Namely, if \(X_{1}, X_{2}, \ldots\) are independent and identically distributed with mean \(\mu\) and variance \(\sigma^{2}\) then, for any \(\varepsilon>0\), $$ P\left\\{\left|\frac{X_{1}+X_{2}+\cdots+X_{n}}{n}-\mu\right|>\varepsilon\right\\} \rightarrow 0 \quad \text { as } n \rightarrow \infty $$

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