Questions: Determine whether the distribution is a discrete probability distribution. x P(x) 0 0.09 1 0.21 2 0.40 3 0.21 4 0.09 Is the distribution a discrete probability distribution? Why? Choose the correct answer below. A. No, because the total probability is not equal to 1. B. No, because some of the probabilities have values greater than 1 or less than 0. C. Yes, because the probabilities sum to 1 and are all between 0 and 1, inclusive. D. Yes, because the distribution is symmetric.

Determine whether the distribution is a discrete probability distribution.
x P(x)
0 0.09
1 0.21
2 0.40
3 0.21
4 0.09

Is the distribution a discrete probability distribution? Why? Choose the correct answer below.
A. No, because the total probability is not equal to 1.
B. No, because some of the probabilities have values greater than 1 or less than 0.
C. Yes, because the probabilities sum to 1 and are all between 0 and 1, inclusive.
D. Yes, because the distribution is symmetric.

Transcript text: Determine whether the distribution is a discrete probability distribution. \begin{tabular}{cc} \hline$x$ & $P(x)$ \\ \hline 0 & 0.09 \\ \hline 1 & 0.21 \\ \hline 2 & 0.40 \\ \hline 3 & 0.21 \\ \hline 4 & 0.09 \\ \hline \end{tabular} Is the distribution a discrete probability distribution? Why? Choose the correct answer below. A. No, because the total probability is not equal to 1. B. No, because some of the probabilities have values greater than 1 or less than 0. C. Yes, because the probabilities sum to 1 and are all between 0 and 1 , inclusive. D. Yes, because the distribution is symmetric.

Solution

Solution Steps

Step 1: Define the Probabilities

The given probabilities are: \[ P(0) = 0.09, \quad P(1) = 0.21, \quad P(2) = 0.40, \quad P(3) = 0.21, \quad P(4) = 0.09 \]

Step 2: Check Individual Probabilities

We need to verify that each probability $ P(x) $ satisfies the condition $ 0 \leq P(x) \leq 1 $:

$ 0.09 $ is between $ 0 $ and $ 1 $
$ 0.21 $ is between $ 0 $ and $ 1 $
$ 0.40 $ is between $ 0 $ and $ 1 $
$ 0.21 $ is between $ 0 $ and $ 1 $
$ 0.09 $ is between $ 0 $ and $ 1 $

All individual probabilities are valid.

Step 3: Calculate the Sum of Probabilities

Next, we calculate the total probability: \[ \text{Total Probability} = P(0) + P(1) + P(2) + P(3) + P(4) = 0.09 + 0.21 + 0.40 + 0.21 + 0.09 = 0.9999999999999999 \]

Step 4: Determine Discrete Probability Distribution

For a distribution to be a discrete probability distribution, the total probability must equal $ 1 $ and all individual probabilities must be between $ 0 $ and $ 1 $.