Question

Let X be a discrete valued random variable with cumulative dist. F(x) is/are correct:

• A. F(x) is a left continuous
• B. always a positive F(x)
• C. has jump discontinuity
• D. is non decreasing F(x)

Hint:

Remember the shape of a discrete CDF: it looks like a staircase. This visual helps recall its properties: it's non-decreasing (stairs go up), it's right-continuous (you land on the step), and it has jumps (the risers of the stairs).

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The question asks to identify the correct properties of the Cumulative Distribution Function (CDF), F(x), for a discrete random variable.
Step 2: Key Properties of a Discrete CDF:
The CDF, F(x) = P(X $\le$ x), for a discrete random variable has the following fundamental properties:
1. Non-decreasing: For any $x_1<x_2$, it must be that $F(x_1) \le F(x_2)$.
2. Range: The value of F(x) is always between 0 and 1, inclusive (i.e., $0 \le F(x) \le 1$).
3. Limits: $\lim_{x\to-\infty} F(x) = 0$ and $\lim_{x\to\infty} F(x) = 1$.
4. Right-continuity: The function is continuous from the right, meaning $\lim_{h\to 0^+} F(x+h) = F(x)$.
5. Step Function: The graph of a discrete CDF is a step function, which is constant between the possible values of the random variable and has jumps at these values. The size of the jump at a point 'k' is equal to the probability P(X=k).
Step 3: Detailed Explanation:
Let's evaluate the given options based on these properties:
- (A) F(x) is a left continuous: This is FALSE. A discrete CDF is right-continuous, not left-continuous. There is a jump as you approach a value from the left.
- (B) always a positive F(x): This is FALSE. F(x) can be equal to 0 for values of x less than the minimum possible value of the random variable. The correct property is that it is non-negative ($F(x) \ge 0$).
- (C) has jump discontinuity: This is TRUE. For a discrete random variable, the CDF increases in jumps at each value that the variable can take with a non-zero probability.
- (D) is non decreasing F(x): This is TRUE. As x increases, the probability P(X $\le$ x) can only increase or stay the same; it can never decrease.
Step 4: Final Answer:
The correct properties from the list are that the CDF has jump discontinuities and is non-decreasing.