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.

Transcript text: 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

Solution Steps

Step 1: Define the Random Variable

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

Step 2: Calculate the Expected Value

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

Step 3: Calculate the Variance

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: \[ \sigma^2 = 19 \cdot 0.4 \cdot 0.6 = 4.56 \]

Step 4: Calculate the Standard Deviation

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