Explain the concepts of Sample Space, Random Variable, and Probability Mass Function with examples.

Here's a summary of the concepts from your notes:

• Sample Space: In a statistical experiment, the set of all possible outcomes is called the Sample Space.

  • Example: For rolling a die, the sample space is $\{1, 2, 3, 4, 5, 6\}$.

• Random Variable: A random variable, often denoted by $X$, is a function that assigns a real number to each outcome in the sample space.

  • A random variable is discrete if it takes on a finite or countably infinite number of integer values.
  • Example: If a fair coin is tossed twice, let $X$ denote the number of heads. The possible values for $X$ are $0, 1, 2$.

• Probability Mass Function (PMF): For a discrete random variable $X$ with possible values $\{x_1, x_2, \dots, x_n\}$, the probability mass function is written as $P(X=x)$, which gives the probability that the random variable $X$ takes on a specific value $x$.

  • Example: For tossing three coins, the sample space is $S = \{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT\}$. If $X$ is the number of heads, the PMF is: $$\begin{array}{|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 \\ \hline P(X=x) & 1/8 & 3/8 & 3/8 & 1/8 \\ \hline \end{array}$$