Questions: Is the following table a valid discrete probability distribution? x 0 1 2 3 4 P(X=x) 0.111 0.214 0.312 0.163 0.159 No, since the probabilities do not add up to 1. No, since not all the probabilities are between 0 and 1. Yes, since all the probabilities are between 0 and 1 and the probabilities add up to one. Yes, since all the x values are greater than 0.

Solution Steps

To determine if the table is a valid discrete probability distribution, we need to check two conditions: (1) all probabilities must be between 0 and 1, and (2) the sum of all probabilities must equal 1.

To determine if the given table is a valid discrete probability distribution, we first check if all probabilities $P(X=x)$ are between 0 and 1. The probabilities given are:

$$P(X=0) = 0.111, \quad P(X=1) = 0.214, \quad P(X=2) = 0.312, \quad P(X=3) = 0.163, \quad P(X=4) = 0.159$$

All these values satisfy $0 \leq P(X=x) \leq 1$.

Next, we calculate the sum of all probabilities:

$$0.111 + 0.214 + 0.312 + 0.163 + 0.159 = 0.959$$

Since the sum is not equal to 1, the condition for a valid probability distribution is not met.

Final Answer