Let (X) have the following CDF:

𝐹
𝑋
(
𝑥
)
=
{
0
,

𝑥

Question

Let (X) have the following CDF:

𝐹
𝑋
(
𝑥
)
=
{
0
,
	
𝑥
<
0
,

1
4
,
	
0
≤
𝑥
<
1
,

1
4
+
𝑥
−
1
2
,
	
1
≤
𝑥
<
2
,

1
,
	
𝑥
≥
2.

Which statement is correct?

BlackTom AI · Accepted Answer

F has an atom at x = 0 of size 1/4 because F(0) − lim_{x→0−}F(x) = 1/4 − 0 = 1/4. Key facts: a CDF is right-continuous and nondecreasing; point mass (atom) at a is F(a) − F(a−); a continuous part has density f(x) = F′(x) where F is differentiable.

Interpreting the middle formula as F(x) = 1/4 + (x − 1)/2 for 1 ≤ x < 2 (the only interpretation that keeps 0 ≤ F ≤ 1), compute the jumps and continuous part:

1) Jump at x = 0:
- F(0−) = 0, F(0) = 1/4, so P(X = 0) = 1/4.

2) Behavior at x = 1:
- From the left, F(1−) = 1/4 (since 0 ≤ x < 1 gives constant 1/4).
- From the right, F(1) = 1/4 + (1 − 1)/2 = 1/4.
- So F is continuous at 1 and there is no atom at x = 1 (P(X = 1) = 0).

3) Continuous density on (1,2):
- For 1 < x < 2, F(x) = 1/4 + (x − 1)/2, derivative f(x) = dF/dx = 1/2.
- Total continuous mass on (1,2) = ∫_{1}^{2} (1/2) dx = (1/2)(2 − 1) = 1/2.

4) Jump at x = 2:
- F(2−) = 1/4 + (2 − 1)/2 = 1/4 + 1/2 = 3/4; F(2) = 1, so P(X = 2) = 1 − 3/4 = 1/4.

Now compute P(1 < X ≤ 1.5):
- F(1.5) = 1/4 + (1.5 − 1)/2 = 1/4 + 0.5/2 = 1/4 + 1/4 = 1/2.
- F(1) = 1/4, so P(1 < X ≤ 1.5) = F(1.5) − F(1) = 1/2 − 1/4 = 1/4.

Why the correct option is right:
- We found discrete masses P(X = 0) = 1/4 and P(X = 2) = 1/4, plus a continuous density on (1,2) with total mass 1/2. That is the definition of a mixed distribution. The statements P(X = 0) = 1/4 and P(1 < X ≤ 1.5) = 1/4 are verified above.

Why other options are incorrect:
- “Purely continuous” is wrong because there are atoms at x = 0 and x = 2.
- “Invalid CDF because flat on [0,1)” is wrong: flat segments are allowed and indicate an atom at the left endpoint.
- “P(X = 1) = 1/4” is incorrect because F is continuous at 1 so P(X = 1) = 0.

Conclusion: the correct statement is “P(X = 0) = 1/4, (P(1 < X ≤ 1.5) = 1/4), and X is mixed,” which matches the provided answer.

Let (X) have the following CDF: 𝐹 𝑋 ( 𝑥 ) = { 0 , 𝑥 < 0 , 1 4 , 0 ≤ 𝑥 < 1 , 1 4 + 𝑥 − 1 2 , 1 ≤ 𝑥 < 2 , 1 , 𝑥 ≥ 2. Which statement is correct?单项选择题

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