Random Variables in Statistics

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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.

Random Experiments: Random Variables in Statistics

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.

Types of Random Variable in Statistics

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 $S$.

Read about Pseudo Random Numbers

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Split Plot Design in Agriculture

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The article is about the use and application of split plot design in Agriculture, here we will discuss the conditions in which split plot design should be used in agriculture, the related real-life examples of split plot design, and the model of the design. In factorial experiments, there are certain situations where it becomes difficult to handle all the combinations of different levels of the factors. This may be because of the following reasons:

  • The nature of the factors may be such that levels of one factor require large experimental units as compared to the levels of other factors. For example, If the two factors are Rowing Methods and Nitrogen Levels”, then in the two-factor experiment the rowing methods require machinery, so they require large experimental units, and the nitrogen levels can be applied to the smaller units.
  • Greater precision may be required for levels of one factor as compared to the levels of other factors. For example, If we want to compare two factors, varieties, and fertilizers, and more precision is required for fertilizers, then varieties would be in the larger units and the fertilizers would be in the smaller units.
  • It may be that new treatments have to be introduced into an experiment that is already in progress.

Conditions in which Split Plot Design Used

The split plot design (and a variation, the split block) is frequently used for factorial experiments in which the nature of the experimental material or the operations involved makes it difficult to handle all factor combinations in the same manner.

  • If irrigation is more difficult to vary on a small scale and fields are large enough to be split, a split-plot design becomes appropriate.
  • Usually used with factorial sets when the assignment of treatments at random can cause difficulties, large-scale machinery can required for one factor but not another irrigation and tillage.
  • Plots that receive the same treatment must be grouped.
  • Degree of Precision: For greater precision for Factor $B$ than for factor $A$, the factor $B$ should be assigned to the subplot and factor $A$ to the main plot.
  • Relative Size of the Main Effects: If the main effect of (say factor $B$) is much larger and easier to detect than that of the other factor (factor $A$), the factor $B$ can be assigned to the main plot, and factor $A$ to the subplot. This increases the chance of detecting the difference among levels of factor $A$ which has a smaller effect.
  • Management Practices: The cultural practices required by a factor may dictate the use of large plots. For example, in an experiment to evaluate water management and variety, it may be desirable to assign water management to the main plot to minimize water movement between adjacent plots, facilitate the simulation of the water level required, and reduce border effects.

Split Plot Design in Agriculture: Irrigation and Fertilizer (Example 1)

In agricultural experiments involving two factors “irrigation” and “nitrogen” fertilizer. Sometimes, it is very convenient to apply different levels of irrigation to small neighbouring plots but there is no such difficulty for the application of different levels of nitrogen fertilizer. To meet such situations, it is desirable to have different sizes of the experimental units in the same experiment. For this purpose, we have two sizes of the experimental units. First, a design with bigger plots is taken to accommodate the factors that require bigger plots. Next, each of the bigger plots is split into as many plots as the number of treatments coming from the other factors.

The bigger plots are called main plots. The treatments allotted to them are called main plot treatments or simply main treatments. The consequent parts of the main plots are called sub-plots or split plots and the treatments allotted to them are called sub-plot treatments. The different types of treatments are allotted at random to their respective plot. Such a design is called split-plot design.

Split Plot design in Agriculture

Split Plot Design in Agriculture: Irrigation and Fertilizer (Example 2)

Let there be 3 levels of irrigation prescribing 3 different amounts of water per plot and 4 doses of nitrogen fertilizer.

First, a randomized block design with a suitable plot is taken with 3 levels of irrigation as treatments say with 5 replications of the design. The irrigation treatments are then allotted at random to each five blocks, each consisting of 4 sub-plots.

Next, each of these main plots is split into 4 sub-plots to accommodate the 4 levels of nitrogen. The main 15 plots serve as 15 replications of the subplot treatments. Treatments are allotted at random to sub-plots of each of the main plots. The split-plot design is the combination of two or more randomized designs depending on several factors, such as the plots of one design from the block of another design. The main plot treatment or the levels of one factor or different factors each of which requires a similar plot size.

Model of Split Plot Design

\begin{align} y_{ijk} &= \mu + \tau_i + \beta_j + (\tau \beta){ij} + \gamma_k + (\tau \gamma){ik} + (\beta\gamma){jk}+(\tau \beta\gamma){ijk} + \varepsilon_{ijk}\\
i &= 1,2,\cdots, a \text{ levels of factor } A\\
j &= 1,2,\cdots, b \text{ levels of factor } B\\
k &= 1,2,\cdots, c \text{ levels of factor } C
\end{align}

Model Terms

  • Linear Terms
    • $\mu$: Overall mean
    • $\tau_i$: Effect of $i$th level of $A$
    • $\beta_j$: Effect of $j$th level of $B$
    • $\gamma_k$: Effect of $k$th level of $C$
  • Interactions Terms
    • $(\tau \beta){ij}$: Interaction effect of $A$ and $B$\ $(\tau \gamma){ik}$: Interaction effect of $A$ and $C$\
    • $(\beta\gamma){jk}$: Interaction effect of $B$ and $C$\ $(\tau\beta\gamma){ijk}$:Interaction effect of $A$, $B$ and $C$ \item \textbf{Error} $\varepsilon{ijk}$: Random error at $i$th level of $A$, $j$th level of $B$ and $k$th level of $C$\
    • $\varepsilon_{ijk} \sim NID(0,\sigma_{\varepsilon}^2)$
  • Response
    • $y_{ijk}$: Response of $i$th level of $A$, $j$th level of $B$ and $k$th level of $C$

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Important MCQs DOE Quiz 4

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The quiz contains MCQs on the Design of Experiments DOE Quiz. Most MCQs on the DOE Quiz are from Basics of Design of Experiments.

Online Multiple Choice Questions about Design of Experiments with Answers

1. Randomized complete block design is used in agriculture when?

 
 
 
 

2. Conducting Bayesian experimentation we use:

 
 
 
 

3. Robustness against missing observations means?

 
 
 
 

4. What is a random experiment?

 
 
 
 

5. What treatments are continuous quantitative variables we should use?

 
 
 
 

6. When prior knowledge of variables is available we should use?

 
 
 
 

7. When treatments are continuous quantitative variables we use?

 
 
 
 

8. Probability theory is based on the paradigm of:

 
 
 
 

9. Evaluation and comparison of basic design configuration is important applications in:

 
 
 
 

10. The first step in the random experiment is:

 
 
 
 

11. The important use of DOE in life sciences is?

 
 
 
 

12. Common types of DOE for environmental sciences include.

 
 
 
 

13. What is the design of the experiment?

 
 
 
 

14. What is the main characteristic of a designed experiment?

 
 
 
 

15. What is the purpose of the experiment?

 
 
 
 

16. The important use of DOE in engineering is?

 
 
 
 

17. Robustness against outliers means?

 
 
 
 

18. When the experiment is to be repeated a large number of times under similar conditions, this is called?

 
 
 
 

19. One of the main objectives of an experiment?

 
 
 
 

20. The most simple blocked design is:

 
 
 
 


Design of experiments (DOE) is a systematic method used to plan, conduct, analyze, and interpret controlled tests to study the relationship between factors and outcomes. Design of Experiment is a powerful tool used in various fields, including science, engineering, and business, to gain insights and optimize processes.

Design of Experiments DOE Quiz

By following the principles of DOE, one can conduct more efficient and informative experiments, ultimately leading to better decision-making and improved outcomes in various fields.

DOE Quiz with Answers

  • What is the purpose of the experiment?
  • What is a random experiment?
  • Probability theory is based on the paradigm of:
  • What is the design of the experiment?
  • What is the main characteristic of a designed experiment?
  • The first step in the random experiment is:
  • One of the main objectives of an experiment?
  • Robustness against missing observations means?
  • Robustness against outliers means?
  • Randomized complete block design is used in agriculture when?
  • When treatments are continuous quantitative variables we use?
  • The most simple blocked design is:
  • The important use of DOE in engineering is?
  • What treatments are continuous quantitative variables we should use?
  • Evaluation and comparison of basic design configuration is important applications in:
  • The important use of DOE in life sciences is?
  • When prior knowledge of variables is available we should use?
  • Conducting Bayesian experimentation we use:
  • Common types of DOE for environmental sciences include.
  • When the experiment is to be repeated a large number of times under similar conditions, this is called?

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Student’s t Table Free Download, 2024

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The t-distribution was discovered by W. S. Gosset and R.A. Fisher. The entries in Student’s t table entries are the critical values (percentiles) for the t distribution. The applications of Student’s t distribution are related to (i) the sampling distribution of the mean $\overline{x}$, (ii) the distribution of a difference $(\overline{x}_1 – \overline{x}_2)$ of two independent populations, (iii) the distribution of two paired (dependent) populations, and (iv) the significance of correlation coefficient. It is also used for constructing confidence intervals for small samples. The Student’s t distribution is a crucial tool in statistical analysis, especially when dealing with small sample sizes. It helps us make informed decisions based on our data, even when the population standard deviation is unknown.

student's t table, Student's t Distribution

The Student’s t variable can be generated by dividing the standard normal random variable ($Z$) with the square root of a $\chi^2_{v}$ random variable. The $\chi^2_v$ is itself divided by its parameter $v$. That is

\begin{align*}
t_v &= \frac{x – \mu }{s_v} = \frac{\tfrac{(x-\mu)}{\sigma} }{\sqrt{\dfrac{\frac{v\times s^2_v}{\sigma^2} } {v}}}\\
&= \frac{Z}{\sqrt{\dfrac{\chi^2_v}{v}}}
\end{align*}

where

PDF of Student’s t Distribution

The PDF of t having $v$ degrees of freedom is

$$p(t_v) = K_v (1+\frac{t^2}{v})^{-\frac{v+1}{2}}$$

where

$$K_v = \frac{\Gamma \left[ \frac{(v+1}{2} \right]}{\sqrt{v\pi} \left(\frac{v}{2}\right) }$$

The t distribution is symmetric about zero and wider than normal density. It has one mode and it tends to be normal as $v\rightarrow \infty$. Note that $\Gamma(x)$ indicates the Gamma function.

Moments of t Distribution

Since the t distribution is symmetric and its PDF is centered at zero, the expectation (average), the median, and the mode are all zero for the t distribution with $v$ degrees of freedom. The variance ($\sigma^2$) equals $\frac{v}{v-2}$ and kurtosis is $\frac{6}{v-4}$.

For bivariate normal population, the distribution of correlation coefficient $r$ is linked with Student’s t distribution through transformation:

$$\frac{r}{\sqrt{\frac{1-r^2}{n-2}}}\rightarrow t_{n-2}$$

Generation of Pseudo Random t Variates

The following algorithm can be used to generate random variates from Student’s $t(v)$ distribution using serially generated independent uniform $U(0,1)$ random variates. For example,

Let $n=v$ (the degrees of freedom)

$C = -2n$

Repeat
$t = 2 \times U(0, 1) – 1$
$u = 2 \times U(0, 1) – 1$
$r = t^2 + u^2$
Until
 $r < 1$
Return
$t \times \sqrt{\frac{n \times (r^C – 1)}{r}}$

Student’s t Table

students-t-table
Students-t-TableDownload

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