Understanding the Random Variable and Probabilities

We are given a biased coin with:

• Probability of Heads, $P(H) = \frac{1}{3}$
• Probability of Tails, $P(T) = \frac{2}{3}$

A random variable X is defined as:

• $X = 1$ if the outcome is Heads
• $X = -1$ if the outcome is Tails

This gives us the probability distribution for X:

• $P(X=1) = P(H) = \frac{1}{3}$
• $P(X=-1) = P(T) = \frac{2}{3}$

Calculating Expected Value E(X)

The expected value, E(X), is calculated as the sum of each possible value of X multiplied by its probability:

$ E(X) = \sum x \cdot P(X=x) $

$ E(X) = (1 \times P(X=1)) + (-1 \times P(X=-1)) $

$ E(X) = \left(1 \times \frac{1}{3}\right) + \left(-1 \times \frac{2}{3}\right) $

$ E(X) = \frac{1}{3} - \frac{2}{3} $

$ E(X) = -\frac{1}{3} $

Calculating Expected Value of X Squared E(X^2)

First, determine the values of $X^2$:

• If $X=1$, then $X^2 = 1^2 = 1$.
• If $X=-1$, then $X^2 = (-1)^2 = 1$.

So, $X^2$ is always 1 in this case. The probability $P(X^2=1)$ is the sum of probabilities for $X=1$ and $X=-1$, which is $1$.

The expected value of $X^2$, E(X^2), is:

$ E(X^2) = \sum x^2 \cdot P(X=x) $

$ E(X^2) = (1^2 \times P(X=1)) + ((-1)^2 \times P(X=-1)) $

$ E(X^2) = (1 \times \frac{1}{3}) + (1 \times \frac{2}{3}) $

$ E(X^2) = \frac{1}{3} + \frac{2}{3} $

$ E(X^2) = 1 $

Variance Calculation Steps

The variance, Var(X), is calculated using the formula:

$ Var(X) = E(X^2) - (E(X))^2 $

Substitute the calculated values:

$ Var(X) = 1 - \left(-\frac{1}{3}\right)^2 $

$ Var(X) = 1 - \frac{1}{9} $

$ Var(X) = \frac{9}{9} - \frac{1}{9} $

$ Var(X) = \frac{8}{9} $

Therefore, the variance of the random variable X is $\frac{8}{9}$.