Question

1. determine whether the distribution shown below is a valid probability distribution.

x	p(x)
3	0.39
4	-0.18
5	0.53

is the distribution valid?

• no, because p(4)<0
• no, because the first value of x is 2, and not 0 or 1.
• yes it is valid because $\sum p(x)=1$
• no, because $\sum p(x)\

eq 1$

1. determine whether the distribution shown below is a valid probability distribution.

x	p(x)
-1	0.26
0	0.27
1	0.27

is the distribution valid?

• no, because of x = -2<0
• no, because $\sum p(x)\

eq 0$

• no, because $\sum p(x)\

eq 1$

• yes it is valid because $\sum p(x)=1$ and each $p(x)>0$

Explanation:

Question 5

Step1: Recall validity conditions

A valid probability distribution requires two conditions: (1) Each probability \( p(x) \geq 0 \) for all \( x \); (2) The sum of all probabilities \( \sum p(x) = 1 \).

Step2: Check each condition

• For \( p(4) = -0.18 \), this is less than 0, violating the first condition.
• Let's also check the sum: \( 0.26 + 0.39 + (-0.18) + 0.53 = 0.26 + 0.39 = 0.65; 0.65 - 0.18 = 0.47; 0.47 + 0.53 = 1 \). But since \( p(4) < 0 \), the first condition fails. The option about \( x \) starting at 2 is irrelevant (probability distributions don't require \( x \) to start at 0 or 1). The sum is 1, but the negative probability makes it invalid. So the correct reason is \( p(4) < 0 \).

Step1: Recall validity conditions

Valid probability distribution: (1) \( p(x) \geq 0 \) for all \( x \); (2) \( \sum p(x) = 1 \).

Step2: Check each condition

• Check \( p(x) \): \( 0.2, 0.26, 0.27, 0.27 \) are all \( > 0 \).
• Check the sum: \( 0.2 + 0.26 + 0.27 + 0.27 = 0.2 + 0.26 = 0.46; 0.46 + 0.27 = 0.73; 0.73 + 0.27 = 1 \). Both conditions are satisfied. The options about \( x = -2 < 0 \) are irrelevant ( \( x \) can be negative in a probability distribution for a discrete random variable, it's the probability that must be non - negative). The sum is 1, and each \( p(x) > 0 \), so it's valid.

Answer:

A. No, because \( p(4) < 0 \)

Question 6