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liq [111]
3 years ago
7

Identify the expression for calculating the mean of a binomial distribution.

Mathematics
1 answer:
Gre4nikov [31]3 years ago
3 0
A random variable X following a binomial distribution with success probability p across n trials has PMF

\mathbb P(X=x)=\begin{cases}\dbinom nxp^x(1-p)^{n-x}&\text{for }x\in\{0,1,\ldots,n\}\\\\0&\text{otherwise}\end{cases}

where \dbinom nx=\dfrac{n!}{x!(n-x)!}.

The mean of the distribution is given by the expected value which is defined by

\mathbb E(X):=\displaystyle\sum_xx\mathbb P(X=x)

where the summation is carried out over the support of X. So the mean is

\displaystyle\sum_{x=0}^nx\binom nxp^x(1-p)^{n-x}

Because this is a proper distribution, you have

\displaystyle\sum_x\mathbb P(X=x)=1

which is a fact that will be used to evaluate the sum above.

\displaystyle\sum_{x=0}^nx\binom nxp^x(1-p)^{n-x}
\displaystyle\sum_{x=1}^nx\binom nxp^x(1-p)^{n-x}
\displaystyle\sum_{x=1}^n\frac{xn!}{x!(n-x)!}p^x(1-p)^{n-x}
\displaystyle np\sum_{x=1}^n\frac{(n-1)!}{(x-1)!(n-x)!}p^{x-1}(1-p)^{n-x}
\displaystyle np\sum_{x=1}^n\frac{(n-1)!}{(x-1)!((n-1)-(x-1))!}p^{x-1}(1-p)^{(n-1)-(x-1)}

Letting y=x-1, this becomes

\displaystyle np\sum_{y=0}^{n-1}\frac{(n-1)!}{y!((n-1)-y)!}p^y(1-p)^{(n-1)-y}

Observe that the remaining sum corresponds to the PMF of a new random variable Y which also follows a binomial distribution with success probability p, but this time across n-1 trials. Therefore the sum evaluates to 1, and you're left with np as the expression for the mean for X.
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