Question

Suppose that X is a random variable that has a binomial uncertainty distribution with parameters n = 10 and π = 0.4. Calculate the numerical value of the probability that X = 6. What are the numerical values of the mean and standard deviation of the uncertainty distribution?

Answer 1

The numerical values of the mean and standard deviation are 4 and 1.55, respectively

The numerical value of the probability that x = 6.

The given parameters are:

n = 10

π = 0.4

The probability is then calculated as:

$P(x) = ^nC_x * \pi^x *(1-\pi)^{n-x}$

So, we have:

$P(6) = ^{10}C_6 * 0.4^6 *(1-0.4)^4$

Apply the combination formula

$P(6) = \frac{10!}{6!4!} * 0.4^6 *0.6^4$

So, we have:

$P(6) = 210 * 0.4^6 *0.6^4$

Evaluate

P(6) = 0.1115

Hence, the numerical value of the probability that x = 6 is 0.1115

The numerical values of the mean and standard deviation

The mean value is:

$\bar x = n\pi$

This gives

$\bar x = 10 * 0.4$

$\bar x = 4$

The standard deviation value is:

$\sigma = \sqrt{\bar x(1-\pi)$

This gives

$\sigma = \sqrt{4(1-0.4)$

$\sigma = 1.55$

Hence, the numerical values of the mean and standard deviation are 4 and 1.55, respectively

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