Question

determine whether the distribution is a probability distribution.

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$$\begin{array}{ccccccc} x & 0 & 1 & 2 & 3 & 4 & 5 \\\\ p(x) & \\frac{1}{20} & \\frac{1}{2} & \\frac{1}{10} & \\frac{3}{5} & \\frac{1}{100} & \\frac{1}{25} \\end{array}$$

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is the probability distribution a discrete distribution? why? choose the correct answer below.

a. no, because some of the probabilities have values greater than 1 or less than 0.
b. yes, because the probabilities sum to 1 and are all between 0 and 1, inclusive.
c. yes, because the distribution is symmetric.
d. no, because the total probability is not equal to 1.

Explanation:

Verify individual probability bounds

$$ 0 \le P(x) \le 1 \quad \text{for all } x \in \{0, 1, 2, 3, 4, 5\} $$

$$ \frac{1}{20}, \frac{1}{2}, \frac{1}{10}, \frac{3}{5}, \frac{1}{100}, \frac{1}{25} \in [0, 1] $$

Calculate the sum of probabilities

$$ \sum P(x) = \frac{1}{20} + \frac{1}{2} + \frac{1}{10} + \frac{3}{5} + \frac{1}{100} + \frac{1}{25} $$

$$ \sum P(x) = \frac{5}{100} + \frac{50}{100} + \frac{10}{100} + \frac{60}{100} + \frac{1}{100} + \frac{4}{100} = \frac{130}{100} = 1.3 $$

Determine if it is a probability distribution

$$ \sum P(x) = 1.3 e 1 $$

Answer:

• A. No, because some of the probabilities have values greater than 1 or less than 0.
• B. Yes, because the probabilities sum to 1 and are all between 0 and 1, inclusive.
• C. Yes, because the distribution is symmetric.
• D. No, because the total probability is not equal to 1. (Correct answer)