Consider the following probability distribution for a discrete random variable $X$: $X$: $1$ $2$ $3$ $4$, $P\left(X\right)$: $0.1$ $0.3$ $0.4$ $0.2$. Which of the following statements is true about this distribution?

• A. The probability of $X=2$ is the highest among all values.
• B. The sum of all probabilities is equal to $1$.
• C. The sum of all probabilities is greater than $1$.
• D. The probability of $X=3$ is less than the probability of $X=1$.

Answer: (B)

Verified step by step guidance

1. Step 1: Understand that for any discrete probability distribution, the sum of all probabilities must equal 1. This is a fundamental property of probability distributions.
2. Step 2: List the given probabilities for each value of X: \(P(X=1) = 0.1\), \(P(X=2) = 0.3\), \(P(X=3) = 0.4\), and \(P(X=4) = 0.2\).
3. Step 3: Calculate the sum of all probabilities by adding them together: \$0.1 + 0.3 + 0.4 + 0.2$.
4. Step 4: Compare the sum obtained in Step 3 to 1 to verify if the sum equals 1, is less than 1, or greater than 1.
5. Step 5: Analyze the individual probabilities to determine which value of X has the highest probability and compare specific probabilities as stated in the options (e.g., compare \(P(X=2)\) and \(P(X=3)\), and \(P(X=3)\) and \(P(X=1)\)).