Consider the following statements regarding random variables : 1. A discrete random variable will have a countable number of distinct values. 2. Continuous random variable is defined for the system which generates a finite number of outcomes within a finite period of time. Which of the above statements is/are correct?
Random Variable Definitions Analysis
We need to evaluate the correctness of two statements regarding random variables.
Statement 1: Discrete Random Variable
- Definition: A discrete random variable is characterized by taking on a finite number of distinct values or a countably infinite number of distinct values.
- Analysis: The statement accurately describes a discrete random variable as having a "countable number of distinct values."
- Conclusion: Statement 1 is correct.
Statement 2: Continuous Random Variable
- Definition: A continuous random variable can take on any value within a given range or interval. The set of possible values is uncountable.
- Analysis: The statement defines a continuous random variable as generating a "finite number of outcomes within a finite period of time." This is inaccurate. Continuous variables deal with uncountable possibilities, not finite outcomes within a timeframe. This description does not align with the standard definition.
- Conclusion: Statement 2 is incorrect.
Final Conclusion
Based on the analysis, only statement 1 is correct. Therefore, the correct option is the one that includes "1 only".
Important Questions from Random Variables
Let $X$ and $Y$ be continuous random variables with probability density functions $P_X(x)$ and $P_Y(y)$, respectively. Further, let $Y = X^2$ and $P_X(x) = \begin{cases} 1, & x\in (0,1] \\ 0, & \text{otherwise} \end{cases}$
Which one of the following options is correct?- Two fair dice (with faces labeled 1, 2, 3, 4, 5, and 6) are rolled. Let the random variable $X$ denote the sum of the outcomes obtained.
The expectation of $X$ is __________ (rounded off to two decimal places). - Let $X = aZ + b$, where $Z$ is a standard normal random variable, and $a, b$ are two unknown constants. It is given that
$E[X] = 1$, $E[(X – E[X])Z] = –2$, $E[(X – E[X])^2] = 4$,
where $E[X]$ denotes the expectation of random variable $X$. The values of $a, b$ are: Let $Y = Z^2$, $Z = \frac{X - \mu}{\sigma}$, where $X$ is a normal random variable with mean $\mu$ and variance $\sigma^2$. The variance of $Y$ is
Consider a discrete random variable X whose probabilities are given below. The standard deviation of the random variable is ________ (round off to one decimal place).
$x_1$ 1 2 3 4 $P(X = x_i)$ 0.3 0.1 0.3 0.3
