Alfred Martin

2022-01-19

What are the mean and standard deviation of a binomial probability distribution with n=10 and $p=\frac{4}{5}$?

### Answer & Explanation

lovagwb

Explanation:
$Mean=np=10×\frac{4}{5}=8$
$SD=\sqrt{npq}$
$q=1-p=1-\frac{4}{5}=\frac{1}{5}$
$SD=\sqrt{10×\frac{4}{5}×\frac{1}{5}}=1.265$

$\frac{4}{5}=0.8$
The mean of the binomial distribution is interpreted as the mean number of successes for the distribution. To find the mean, use the formula
$\mu =n\cdot p$
where n is the number of trials and p is the probability of success on a single trial. Substituting values for this problem, we have
$\mu =10\cdot 0.8$
Multiplying the expression we have
$\mu =8$
The standard deviation of the binomial distribution is interpreted as the standard deviation of the number of successes for the distribution. To find the standard deviation, use the formula
$\sigma =\sqrt{n\cdot p\cdot \left(1-p\right)}$
where n is the umber of trials and p is the probability of success on a single trial. Substituting values fo this problem, we have
$\sigma =\sqrt{10\cdot 0.8\cdot \left(1-0.8\right)}$
Evaluating the expression on the right, we have
$\sigma =\sqrt{1.6}$
$\sigma =1.26$

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