State True or False: cov(`x,x`)= variance of `x`

Answer

To determine whether the statement "cov(x, x) = variance of x" is true or false, we can follow these steps:

### Step 1: Understand the definition of covariance
Covariance between two random variables \(X\) and \(Y\) is defined as:
\[ \text{cov}(X, Y) = E[XY] - E[X]E[Y] \]
where \(E\) denotes the expected value.

**Hint:** Recall that covariance measures how two variables change together.

### Step 2: Substitute \(Y\) with \(X\) in the covariance formula
Now, if we want to find the covariance of \(X\) with itself, we substitute \(Y\) with \(X\):
\[ \text{cov}(X, X) = E[X \cdot X] - E[X]E[X] \]

**Hint:** Think about what happens when you multiply a variable by itself.

### Step 3: Simplify the expression
This simplifies to:
\[ \text{cov}(X, X) = E[X^2] - (E[X])^2 \]

**Hint:** Recognize that \(E[X^2]\) is the expected value of the square of \(X\).

### Step 4: Recognize the definition of variance
The variance of \(X\) is defined as:
\[ \text{Var}(X) = E[X^2] - (E[X])^2 \]

**Hint:** Variance quantifies the spread of a set of values around their mean.

### Step 5: Compare the two expressions
From the above steps, we see that:
\[ \text{cov}(X, X) = E[X^2] - (E[X])^2 = \text{Var}(X) \]

**Hint:** Look closely at the structure of both equations to see if they match.

### Conclusion
Since we have shown that \(\text{cov}(X, X)\) is equal to \(\text{Var}(X)\), we can conclude that the statement is **True**.

**Final Answer:** True