Random Variables

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.

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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The post is about Random Variable MCQ Questions. There are 20 multiple-choice questions in the quiz. The quiz covers topics related to the generation of random numbers, types of random variables, examples of random variables, Expected value and variance of random variables, and distribution of random variables. Let us start with Random Variable MCQ questions.

Please go to Important Random Variable MCQ Questions 4 to view the test

Online Random Variable MCQ Questions

• Random numbers are generated by some
• In generating random numbers there are ———- number of assumptions to follow
• In generating random numbers the probability of each digit/number is
• In a family with two children, how many are girls there
• A discrete variable is also called
• Discrete data is usually generated by the process
• The sum of dots when two dice are rolled is an example of
• The number of deaths in a road accident is an example of ———- variable
• The number of children in a family is an example of ———- variable
• A variable which can assume all values in the range is called
• Usually, measurements give rise to ———– data
• Continuous variables can assume ———– values
• The amount of milk produced by a cow is ———- variable
• A variable that takes measurable values is called a
• The set of all possible outcomes of a random experiment is called
• A Chi-Square random variable can assume the value
• The number of automobile accidents per year in Multan city is an example of
• Which of the following is a characteristic of the probability distribution of a random variable?
• If $X$ and $Y$ are random variables, then $E(X-Y)$ is equal to
• If $X$ and $Y$ are independent random variables, then $Var(X-Y)$ is equal to

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