Random Variables

Online Random Variable MCQs Test with Answers. Test your knowledge with 20 multiple-choice questions on discrete random variables, expected value (E(X)), variance (Var(X)), standard deviation, and independent random variables. Perfect for probability and statistics students, exam preparation, and data science interviews. Practice key formulas, properties of expectation and variance, and definitions of discrete vs. continuous random variables. Sharpen your skills for college exams, competitive tests, and statistics certifications—free online quiz included! Let us start with the Online Random Variable MCQs Test with Answers.

Online Random Variable MCQs Test with Answers

• If $X$ and $Y$ are random variables with $E(X)=5$ and $E(Y)=8$ then $E(2X+3Y)$ is
• The mean of a discrete random variable is the mean of its
• The mean of a discrete random variable is also called its
• The mean of a discrete random variable is obtained by using the formula
• If $X$ and $Y$ are any random variables, which of the following identities is not true?
• The standard deviation of a discrete random variable is the standard deviation of its
• If $X$ and $Y$ are independent random variables, which of the following identities is always true?
• A random variable that assumes whole numbers and a finite number of values is called
• A random variable that can assume all possible values within a range is called
• Two random variable assuming only a finite number of values is called
• A random variable assuming only a finite number of values is called
• A random variable is also called
• A variable whose value is determined by the outcome of a random experiment is called
• The sum of probabilities of a discrete random variable is
• While tossing 3 coins, the values that a random variable (number of heads) can take
• If $Var(X) = 4$ then $Var(3X + 5)$ is equal to
• If $X$ is a random variable, then $Var(2-3X)$ is equal to
• If $X$ and $Y$ are independent then $Var(X-Y)$
• $Var(X/3)=?$
• $Var(X + 4)=?$

General Knowledge Quizzes

Introduction

A discrete random variable $x$ is said to have a binomial distribution if $x$ (binomial random variable) satisfies the following conditions:

• An experiment is repeated for a fixed number of trials $n$.
• All the trials of the experiments are independent of each other.
• All possible outcomes for each trial of the experiment can be classified into two mutually (complementary) events: one is $S$ (called success) and the other is $F$ (called failure).
• The probability of success $P(S)$ has a constant value of $p$ for every trial (that is, the probability of success is fixed for each trial) and hence the probability of failure $P(F)$ has a constant/fixed value of $q$ for every trial, where $q=1-p$.
• The random variable $x$ counts the number of trials on which $S$ (success) occurred.

Calculating Probabilities for a Binomial Random Variable

If $X$ is a binomial random variable with $n$ trials, probability of success $p$ (and probability of failure $q$), then by the fundamental counting principle, the probability of any outcome in which there are $x$ successes (and therefore $n-x$ failures) is

To count the number of outcomes with $x$ successes and $n-x$ failures, one can observe that the $x$ successes could occur on any $x$ of the $n$ trials. The number of ways of choosing/selecting $x$ trials out of $n$ is $\binom{n}{x}$, so the probability of $x$ successes becomes:

$$P(X=x)=\binom{n}{x} p^x q^{n-x}$$

Example of Binomial Random Experiments

Example: Consider the experiment of flipping a coin 5 times. Let the event of getting Tails on a flip is considered a “success”. Also, suppose that the random variable $T$ is the number of tails obtained, the $T$ will be binomially distribution with $n=5, p=\frac{1}{2}$, and $q=\frac{1}{2}$.

Solution:
Suppose the random variable $T$ represents the number of trials when a coin is flipped three times.
$$P(X=2) = \binom{3}{2}\left(\frac{1}{2}\right)^2 \left(\frac{1}{2}\right)^1 = 0.375$$

Properties of Binomial Distributions

In many cases, one may be interested in the mean and standard deviation of the binomial random variable. If $x$ is a binomial random variable with $n$ trials with probability of success $p$ and probability of failure $q$, then the mean and standard deviation of $x$ can be computed as

• Mean: $E(X) = \mu(x) = np$
• Standard Deviation: $\sigma(x) = \sqrt{npq}$
• Variance: $npq$

Note that

• A binomial distribution is symmetric if $p=q$,
• left skewed if $p>q$ and
• right-skewed if $p<q$

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The post is about the Random Variables MCQs Test. There are 20 multiple-choice questions covering topics about the basics of random variables, types of random variables, dependent and independent variables, expectation and variance of random variables, and joint density of random variables. Let us start with the Random Variables MCQs Quiz.

Please go to Important Random Variables MCQs 5 to view the test

Random Variables MCQs

• $E(X-\mu)$ = ?
• If $E(X) = 1.6$ then $E(5X+10) =$?
• If $X$ is a random variable then $E(4X + 2)$ is
• If $E(X) = 3$ then what will be $E(a+X)$, where $a$ is a constant whose value is 2
• If $P(X=10) = \frac{1}{10}$, then $E(X)=$?
• If $X$ and $Y$ are random variables then $E(X-Y)$ is
• If $X$ and $Y$ are two independent variables then $E(XY)=$
• If $X$ and $Y$ are independent random variables then $SD(X-Y)$ will be
• If $X$ is a random variable and $a$ and $b$ are constants, then $SD(a-bX)$ is
• If $a$ is constant then $Var(a)$ is
• If $Var(X) = 8$ then $Var(X+3)= $
• If $Var(X) = 5$ then $Var(2X + 5) $ =
• If $Var(X) = 2$ and $Var(Y) = 5$, and if $X$ and $Y$ are independent variables, then $Var(2X – Y)=$?
• If $X_1, X_2, \cdots, X_n$ is the joint density of $n$ random variables, say $f(X_1, X_2, \cdots, X_n;\theta)$ which is considered to be a function of $\theta$. Then $L(\theta; X_1,X_2,\cdots,X_n)$ is called:
• A discrete variable is a variable that can assume
• If $X$ is a discrete random variable then the function $f(x)$ is
• If $X$ and $Y$ are random variable then $E(X+y)$ is
• If $X$ and $Y$ are independent random variables then $E(XY)$ is
• If $X$ and $Y$ are independent random variables then $Var(x-y)$ is
• If $X$ is a random variable then $Var(2-3X)$ is

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