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
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 :
The density function of a random variable is given by \(p(x)=Ke^{-\frac{x^{2}}{2}}\) for \(-\infty \lt x \lt \infty\). The value of K should be :
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 - Consider a discrete random variable X that takes values from the set
$S = \{0, 1, 2, 3\}$, being the number of individuals of a species within a habitat.
Consider the probability distribution of X with $Pr(X = 0) = 0.15$,
$Pr(X = 1) = 0.25$ and $Pr(X = 3) = 0.5$, where Pr denotes probability. The value of
$Pr(X = 2)$ is __________. (Round off to two decimal places)
