Define Expectation of a Random Variable.

1. Expectation of a Constant: If $c$ is a constant, then $E[c] = c$.
2. Linearity (Constant Multiple): If $c$ is a constant and $X$ is a random variable, then $E[cX] = cE[X]$.
3. Linearity (Sum/Difference): If $X$ and $Y$ are random variables, then $E[X \pm Y] = E[X] \pm E[Y]$.
4. Expectation of a Sum of Independent Random Variables: If $X_1, X_2, \dots, X_n$ are random variables, then $E[\sum_{i=1}^{n} X_i] = \sum_{i=1}^{n} E[X_i]$. This property holds even if the random variables are not independent.
5. Expectation of a Product of Independent Random Variables: If $X$ and $Y$ are independent random variables, then $E[XY] = E[X]E[Y]$.

c) Let $S$ be the sample space for tossing a fair coin twice. The possible outcomes are:

Each outcome has a probability of $\frac{1}{4}$ since the coin is fair. The random variable $X$ is defined such that $X(e) = 0$ if $e$ is T (tail) and $X(e) = 1$ if $e$ is H (head). This means $X$ counts the number of heads in the two tosses. Let's map the outcomes to the values of $X$:

• $HH \implies X = 1+1 = 2$
• $HT \implies X = 1+0 = 1$
• $TH \implies X = 0+1 = 1$
• $TT \implies X = 0+0 = 0$

The possible values for $X$ are $\{0, 1, 2\}$.
Now, we find the probability for each value of $X$:

• $P(X=0)$: Occurs for outcome $TT$. So, $P(X=0) = \frac{1}{4}$.
• $P(X=1)$: Occurs for outcomes $HT$ and $TH$. So, $P(X=1) = \frac{1}{4} + \frac{1}{4} = \frac{2}{4} = \frac{1}{2}$.
• $P(X=2)$: Occurs for outcome $HH$. So, $P(X=2) = \frac{1}{4}$.

The Probability Distribution Table for $X$ is:

Now, we calculate the Expected Value of $X$, $E[X]$:

The Expected Value of $X$ is $\boxed{1}$.