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
If the odds in favour of any random event A are 5 ∶ 6, then the odds against the event are:
If random variable X follows binomial distribution with parameter n and p with mean 15 and variance 10, then the value of mode is
- Let $X$, $N$, $Y$ and $Z$ be random variables. The variables $X$ and $N$ are independent of each other. $X$ is uniformly distributed between -1 and 1; $N$ follows Normal distribution with zero mean and unity variance.
$Y$ and $Z$ are defined as, $Y = X + N$ and $Z = X^2 + N$.
Which of the following pairs represents the values of correlation between $X$ and $Y$ and that between $X$ and $Z$? Consider the following probability density function for a random variable x.
(a) \(f_{1}(x)=1\ ;\ -1\le x\le 1\)
(b) \(f_{2}(x)=x\ ;\ 0\le x\le 1\)
(c) \(f_{3}(x)=(1-x)\ ;\ -1\le x\le 1\)
Arrange the above functions in terms of the increasing value of mean of random variable x.
Considering all the symbols with their usual meanings, match the following :
List - I List - II (a) \(f_{X}(x)\) (i) \(\displaystyle\int_{-\infty}^{\infty}f_{X,Y}(x,y)\,dx\) (b) \(\displaystyle\int_{-\infty}^{\infty}f_{X}(x)\,dx\) (ii) \(\displaystyle\int_{-\infty}^{a}f_{X}(x)\,dx\) (c) \(F_{X}(a)\) (iii) \(\displaystyle\int_{-\infty}^{\infty}f_{X,Y}(x,y)\,dy\) (d) \(f_{Y}(y)\) (iv) 1 Codes :
