Explain why each of the following is or is not a valid...

Explain why each of the following is or is not a valid probability distribution for a discrete random variable $x$ : $$ \begin{aligned} &\text { a. }\\ &\begin{array}{l|llll} \hline x & 0 & 1 & 2 & 3 \\ p(x) & .2 & .3 & .3 & .2 \\ \hline \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { b. }\\ &\begin{array}{l|lll} \hline x & -2 & -1 & 0 \\ p(x) & .25 & .50 & .20 \\ \hline \end{array} \end{aligned} $$ $$ \begin{aligned} &\mathbf{c.}\\ &\begin{array}{l|rrr} \hline x & 4 & 9 & 20 \\ p(x) & -.3 & 1.0 & .3 \\ \hline \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { d. }\\ &\begin{array}{l|rrrr} \hline x & 2 & 3 & 5 & 6 \\ p(x) & .15 & .20 & .40 & .35 \\ \hline \end{array} \end{aligned} $$

Explain why each of the following is or is not a valid probability distribution for a discrete random variable $x$ :
$$
\begin{aligned}
&\text { a. }\\
&\begin{array}{l|llll}
\hline x & 0 & 1 & 2 & 3 \\
p(x) & .2 & .3 & .3 & .2 \\
\hline
\end{array}
\end{aligned}
$$
$$
\begin{aligned}
&\text { b. }\\
&\begin{array}{l|lll}
\hline x & -2 & -1 & 0 \\
p(x) & .25 & .50 & .20 \\
\hline
\end{array}
\end{aligned}
$$
$$
\begin{aligned}
&\mathbf{c.}\\
&\begin{array}{l|rrr}
\hline x & 4 & 9 & 20 \\
p(x) & -.3 & 1.0 & .3 \\
\hline
\end{array}
\end{aligned}
$$
$$
\begin{aligned}
&\text { d. }\\
&\begin{array}{l|rrrr}
\hline x & 2 & 3 & 5 & 6 \\
p(x) & .15 & .20 & .40 & .35 \\
\hline
\end{array}
\end{aligned}
$$

Key Concepts

Discrete Random Variable

A discrete random variable is a type of random variable that can take on only a countable number of distinct values. This concept is fundamental in probability theory and is often used when dealing with outcomes that are naturally separate and distinct.

Probability Distribution

A probability distribution for a discrete random variable assigns a probability to each possible outcome in such a way that the probabilities are well-defined and consistent with the rules of probability. It is a comprehensive specification of how probabilities are distributed among the outcomes.

Non-negative Probabilities

An essential requirement for any probability distribution is that every probability value must be non-negative. This means that for each possible outcome, the assigned probability must be zero or a positive number, ensuring that probabilities do not contradict the concept of likelihood.

Total Probability Equals One

Another key condition for a valid probability distribution is that the sum of the probabilities for all possible outcomes must equal one. This reflects the certainty that one of the possible outcomes will occur and ensures that the distribution is complete.

Probability Mass Function (PMF)

The probability mass function (PMF) is a function that gives the probability that a discrete random variable is exactly equal to some value. It encapsulates the characteristics of the probability distribution for discrete variables, including the requirements of non-negativity and total probability equaling one.

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