Questions: The random variable X is a binomial random variable with n=19 and p=0.4. What is the expected value of X? Do not round your answer.

Solution Steps

The random variable $X$ follows a binomial distribution with parameters $n = 19$ (number of trials) and $p = 0.4$ (probability of success).

The expected value (mean) of a binomial random variable is given by the formula: $$\mu = n \cdot p$$ Substituting the values: $$\mu = 19 \cdot 0.4 = 7.6$$

The variance of a binomial random variable is calculated using the formula: $$\sigma^2 = n \cdot p \cdot q$$ where $q = 1 - p$. Thus, we have: $$q = 1 - 0.4 = 0.6$$ Now substituting the values: $$\sigma^2 = 19 \cdot 0.4 \cdot 0.6 = 4.56$$

The standard deviation is the square root of the variance: $$\sigma = \sqrt{n \cdot p \cdot q} = \sqrt{19 \cdot 0.4 \cdot 0.6} \approx 2.1354$$

Final Answer