For Bernoulli Distribution , state formula for E(X) and V(X)

Answer

To find the expected value \( E(X) \) and variance \( V(X) \) for a Bernoulli distribution, we follow these steps:

### Step 1: Define the Bernoulli Distribution
A Bernoulli distribution is a discrete probability distribution for a random variable which takes the value 1 with probability \( p \) (success) and the value 0 with probability \( 1 - p \) (failure).

### Step 2: Calculate the Expected Value \( E(X) \)
The expected value \( E(X) \) of a Bernoulli random variable can be calculated using the formula:
\[ E(X) = \sum_{x} x \cdot P(X = x) \]
For a Bernoulli distribution, this becomes:
\[ E(X) = 1 \cdot P(X = 1) + 0 \cdot P(X = 0) = 1 \cdot p + 0 \cdot (1 - p) = p \]

### Step 3: Calculate the Variance \( V(X) \)
The variance \( V(X) \) of a Bernoulli random variable can be calculated using the formula:
\[ V(X) = E(X^2) - (E(X))^2 \]
For a Bernoulli distribution, since \( X \) can only take values 0 and 1:
\[ E(X^2) = 1^2 \cdot P(X = 1) + 0^2 \cdot P(X = 0) = 1 \cdot p + 0 \cdot (1 - p) = p \]
Thus, we can substitute this into the variance formula:
\[ V(X) = E(X^2) - (E(X))^2 = p - p^2 \]
This can be simplified to:
\[ V(X) = p(1 - p) \]

### Final Formulas
So, the formulas for the expected value and variance of a Bernoulli distribution are:
\[ E(X) = p \]
\[ V(X) = p(1 - p) \]