Question

If a random variable $X$ has the following probability distribution, then its variance is nearly: 

• A. $2.8875$
• B. $2.9875$
• C. $2.7865$
• D. $2.785$

Hint:

To find the variance of a probability distribution, compute $E(X^2)$ and use $\text{Var}(X) = E(X^2) - (E(X))^2$.

Answer 1

The Correct Option is A

Step 1: Verify that probabilities sum to 1
We have: $$0.05 + 0.1 + 2K + 0 + 0.3 + K + 0.1 = 1.$$ Solving for $K$: $$2K + K = 1 - (0.05 + 0.1 + 0.3 + 0.1) = 1 - 0.55 = 0.45.$$ $$3K = 0.45 \Rightarrow K = 0.15.$$ Step 2: Compute Expectation $E(X)$
$$E(X) = \sum x P(X=x).$$ $$= (-3)(0.05) + (-2)(0.1) + (-1)(2K) + (0)(0) + (1)(0.3) + (2)(K) + (3)(0.1).$$ $$= (-0.15) + (-0.2) + (-0.3) + 0 + 0.3 + 0.3 + 0.3 = 0.$$ Step 3: Compute $E(X^2)$
$$E(X^2) = \sum x^2 P(X=x).$$ $$= (-3)^2(0.05) + (-2)^2(0.1) + (-1)^2(2K) + (0)^2(0) + (1)^2(0.3) + (2)^2(K) + (3)^2(0.1).$$ $$= (9)(0.05) + (4)(0.1) + (1)(0.3) + 0 + (1)(0.3) + (4)(0.15) + (9)(0.1).$$ $$= 0.45 + 0.4 + 0.3 + 0.3 + 0.6 + 0.9 = 2.8875.$$ Step 4: Compute Variance $\sigma^2(X)$
$$\text{Var}(X) = E(X^2) - (E(X))^2.$$ Since $E(X) = 0$, $$\text{Var}(X) = 2.8875 - 0^2 = 2.8875.$$ Step 5: Conclusion
Thus, the final answer is: $$\boxed{2.8875}.$$