Random Variables

 The post is about MCQ Random Variables. There are 20 multiple-choice questions related to random experiments, random variables and types of random variables, expectations, discrete random variables, and continuous random variables. Let us start with the MCQ Random Variables Quiz.

MCQ Random Variables Quiz

•  If $X$ is a continuous random variable, then function $f(X)$ is
• A variable (Random Variable) assuming an infinite number of values is called
• A variable whose value is determined by the outcome of a random experiment is called
• If $X$ and $Y$ are random variable then $E(X + Y)$ is equal to
• If $X$ is a discrete random variable, the function $f(X)$ is
• When four coins are tossed, the value of a random variable (Numbers of head) is
• A variable (Random Variable) assuming a finite number of values is called
• If $X$ and $Y$ are independent random variables then $E(XY)$ is equal to
• Two random variables $X$ and $Y$ are said to be independent if:
• If $X$ and $Y$ are two independent variables, then
• A continuous random variable is a random variable that can
• If $X$ is a random variable that can take only non-negative values, then
• For a random variable $X$, $E(X)$ is
• If $C$ is a non-random variable, the $E(C)$ is
• A continuous variable is a variable that can assume
• A ———– random variable has a countable number of possible values.
• Which of the following statements accurately describes a key difference between discrete and continuous random variables?
• Which of the following are examples of discrete random variables?
• Which of the following statements describes continuous random variables?
• If $X$ is a random variable and $a$ and $b$ are constants then $Var(aX+ b)$ is equal to

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Introduction to a Random Variable in Statistics

A random variable in statistics is a variable whose value depends on the outcome of a probability experiment. As in algebra, random variables are represented by letters such as $X$, $Y$, and $Z$. A random variable in statistics is a variable whose value is determined by chance. A random variable is a function that maps outcomes to numbers. Read more about random variables in Statistics: Random Variable.

Random Variable in Statistics: Some Examples

• T = the number of tails when a coin is flipped 3 times.
• s = the sum of the values showing when two dice are rolled.
• h = the height of a woman chosen at random from a group.
• V = the liquid volume of soda in a can marked 12 oz.
• W = The weight of an infant chosen at random in a hospital.

Key Characteristics of a Random Variable

• Randomness: The value of a random variable is determined by chance.
• Numerical: It assigns numbers to outcomes.
• Function: It is technically a function that maps outcomes to numbers.

Types of Random Variables

There are two basic types of random variables.

Discrete Random Variables: A discrete random variable can take on only a countable number of values. It can have a finite or countable number of possible values.

Continuous Random Variables: A continuous random variable Can take on any value within a specified interval. It can take on any value in some interval.

Examples of Discrete and Continuous Random Variables

• The variables $T$ and $s$ from above are discrete random variables
• The variables $h$, $V$, and $W$ from above are continuous random variables.

Importance of Random Variables in Statistics

Random variables are fundamental to statistics. Random variables allow us to:

• Use mathematical tools to analyze uncertain events.
• Model the real-world phenomena.
• Calculate probabilities of events.
• Compute expected values and variances.
• Make statistical inferences.

Random variables form the basis for probability distributions and are fundamental to statistical inference. Random variables provide a bridge between the real world of uncertainty and the mathematical world of probability.

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R Frequently Asked Questions

In any experiment of chance, the outcomes occur randomly. For example, rolling a single die is an experiment: Any of the six possible outcomes can occur. Some experiments result in outcomes that are quantitative (such as dollars, weight, or number of children), and others result in qualitative outcomes (such as color or religious preferences). Therefore, random variables in statistics are variables whose value depends on the output of a random experiment.

A random variable is a mathematical abstraction that allows one to assign numerical values to the random variable associated with a probability to indicate the chance of a particular outcome.

Random Experiment

In the random experiment, a numerical value say 0, 1, 2, is assigned to each sample point. Such a numerical quantity whose value is determined by the outcomes of an experiment of chances is known as a random variable (or stochastic variable). Therefore, a random experiment is a process that has a well-defined set of possible outcomes, however, the outcomes for any given trial of the random experiment cannot be predicted in advance. Examples of random experiments are: rolling a die, flipping a coin, and measuring the height of students walking into a class.

Classification of Random Variables in Statistics

A random variable can be classified into a discrete random variable and a continuous random variable.

Discrete Random Variable

A discrete random variable can assume only a certain number of separated values. The discrete random variables can take only finite or countably infinite numbers of distinct values. For example, the Bank counts the number of credit cards carried by a group of customers. The other examples of discrete random variables are: (i) The number of successes in a 5-coin flip experiment, (ii) the number of customers arriving in a store during a specific hour, (iii) the number of students in a class, and (iv) the number of phone calls in a certain day.

Continuous Random Variable

The continuous random variable can assume any value within a specific interval. For example, the width of the room, the height of a person, the pressure in an automobile tire, or the CGPA obtained, etc. The continuous random variable assumes an infinitely large number of values, within certain limitations. For example, the tire pressure measured in pounds per square inch (psi) in most passenger cars might be 32.78psi, 31.32psi, 33.07psi, and so on (any value between 28 and 35). The random variable is the tire pressure, which is continuous in this case.

Definition: A random variable is a real-valued function that takes a defined value for every point in the sample space.

In most of the practical problems, discrete random variables represent count or enumeration data such as the number of books on a shelf, the number of cars crossing a bridge on a certain day or time, or the number of defective items in a production (or a lot). On the other hand, continuous random variables usually represent measurement data such as height, weight, distance, or temperature.

Note: A random variable represents the particular outcome of an experiment, while a probability distribution reports all the possible outcomes as well as the corresponding probability.

Importance of Random Variables

The importance of random variables cannot be ignored, because random variables are fundamental building blocks in the field of probability and statistics. The random variables allow us to:

• Quantify Uncertainty: Since numerical values are assigned to outcomes from a random experiment, one can use mathematical tools such as probability distributions to compute and analyze the likelihood of different events occurring.
• Statistical Analysis: Random variables are essential for performing various types of statistical analyses such as computing expected values, and variance, conducting hypothesis testing, and computing relationships between variables, etc.
• Modeling Real-World Phenomena: One can use random variables to model real-world phenomena with inherent randomness, allowing for predictions and simulations.

Note that each possible outcome of a random experiment is called a sample point. The collection of all possible sample points is called sample space, represented by $S$.

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A Random Variable (random quantity or stochastic variable) is a set of possible values from a random experiment. The domain of a random variable is called sample space. For example, in the case of a coin toss experiment, there are only two possible outcomes, namely heads or tails. A random variable can be either discrete or continuous. The discrete random variable takes only certain values such as 1, 2, 3, etc., and a continuous random variable can take any value within a range such as the height of persons.

By using random variables, one can use the tools of probability and statistics to analyze the outcomes of the experiment. One can calculate such as the probability of getting a certain result, the average outcome, or how spread out the results are.

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