Probability Distribution Discrete Random Variable

A probability distribution for a discrete random variable $X$ is a list of each possible value for $X$ with the probability that $X$ will have that value when the experiment is run. The likelihood for the probability distribution of a discrete random variable is denoted by $P(X=x)$. The probability distribution of a discrete random variable is also called a discrete probability distribution.

A discrete probability distribution is a mathematical function that assigns probabilities to each possible value of a discrete random variable.

Example of Probability Distribution of a Discrete Random Variable

Let $X$ be a random variable representing the number of trials obtained when a coin is flipped three times in an experiment. The sample space of the experiment is:

$$HHH, HHT, HTH, THH, HTT, TTH, THT, TTT$$

where $T$ represents the occurrence of Tail and $H$ represents the occurrence of Head in the above experiment.

Then $X$ has 4 possible values: $0, 1, 2, 3$ for the occurrence of head or tail. The probability distribution for $X$ is given as below:

$X$$P(X)$
0$\frac{1}{8}$
1$\frac{3}{8}$
2$\frac{3}{8}$
3$\frac{1}{8}$
Total$1.0$

In a statistics class of 25 students are given a 5-point quiz. 3 students scored 0; 1 student scored 1, 4 students scored 2, 8 students scored 3, 6 students scored 4, and 3 students scored 5. If a student is chosen at random, and the random variable $S$ is the student’s Quiz Score then the discrete probability distribution of $S$ is

$S$$P(S)$
00.12
10.04
20.16
30.32
40.24
50.12
Total1.0

Note that for any discrete random variable $X$, $0\le P(X) \le 1$ and $\Sigma P(X) =1$.

Finding Probabilities from a Discrete Probability Distribution

Since a random variable can only take one value at a time, the events of a variable assuming two different values are always mutually exclusive. The probability of the variable taking on any number of different values can thus be found by simply adding the appropriate probabilities.

discrete and continuous probability distributions, discrete random variable

Mean or Expected Value of a Discrete Random Variable

The mean or expected value of a random variable $X$ is the average value that one should expect for $X$ over many trials of the experiment in the long run. The general notation of the mean or expected value of a random variable $X$ is represented as

$$\mu_x\quad \text{ or } E[X]$$

The mean of a discrete random variable is computed using the formula

$$E[X]=\mu_x = \Sigma x\cdot P(X)$$

Example 1

From the above experiment of three Coins the Expected value of the random variable $X$ is

$X$$P(X)$$x.P(X)$
0$\frac{1}{8}$$0 \times \frac{1}{8} = 0$
1$\frac{3}{8}$$1 \times \frac{3}{8} = \frac{3}{8}$
2$\frac{3}{8}$$2 \times \frac{3}{8} = \frac{6}{8}$
3$\frac{1}{8}$$3 \times \frac{1}{8} = \frac{3}{8}$
Total$1.0$$\frac{3}{2} = 1.5$

Thus if three coins are flipped a large number of times, one should expect the average number of trials (per 3 flips) to be about 1.5.

Discrete Random Variable, discrete probability distributions

Example 2

Similarly, the mean of the random variable $S$ from the above example is

$S$$P(S)$$S\cdot P(S)$
00.12$0 \times 0.12 = 0$
10.04$1 \times 0.04 = 0.04$
20.16$2 \times 0.16 = 0.32$
30.32$3 \times 0.32 = 0.96$
40.24$4\times 0.24 = 0.96$
50.12$5 \times 0.12 = 0.60$
Total$1.0$$2.88$

Note that $2.88$ is the class average on the statistics quiz as well.

Variance and Standard Deviation of a Random Variable

One may be interested to find how much the values of a random variable differ from trial to trial. To measure this, one can define the variance and standard deviation for a random variable $X$. The variance of $X$ random variable is denoted by $\sigma^2_x$ while the standard deviation of the random variable $X$ is just the square root of $\sigma^2_x$. The formulas of variance and standard deviation of a random variable $X$ are:

\begin{align*}
\sigma^2_x &= \Sigma (x – \mu)^2 P(X)\\
\sigma_x &= \sqrt{\Sigma (x – \mu)^2 P(X)}
\end{align*}

Note that the standard deviation estimates the average difference between a value of $x$ and the expected value.

Calculating the Variance and Standard Deviation

The calculation of standard deviation for a random variable is similar to the calculation of weighted standard deviation in a frequency table. The $P(x)$ can be thought of as the relative frequency of $x$. The computation of variance and standard deviation of a random variable $X$ can be made using the following steps:

  1. Compute $\mu_X$ (mean of the random variable)
  2. Subtract the mean/average from each of the possible values of $X$. These values are called the deviations of the $X$ values.
  3. Square each of the deviations calculated in the previous step.
  4. Multiply each squared deviation (calculated in step 3) by the corresponding probability $P(x)$.
  5. Sum the results of step 4. The variance of the random variable will be obtained representing $\sigma^2_X$.
  6. Take the square root of the $\sigma^2_X$ computed in Step 5.

Importance of Discrete Probability Distributions

  • Modeling Real-World Phenomena: Discrete Distributions help us understand and model random events in various fields of life such as engineering, finance, and the sciences.
  • Decision Making: These distributions provide a framework for making informed decisions under uncertainty.
  • Statistical Inference: These are used to make inferences about populations based on sample data.

FAQs about the Probability Distribution of a Discrete Random Variable

  1. Define the probability distribution.
  2. What is a random variable?
  3. What is meant by an expected value or a random variable?
  4. What is meant by the variance and standard deviation of a random variable?

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The t Distribution 2024

Introduction to t Distribution

The Student’s t distribution or simply t distribution is a probability distribution similar to the normal probability distribution with heavier tails. The t distribution produces values that fall far from the average compared to the normal distribution. The t distribution is an important statistical tool for making inferences about the population parameters when the population standard deviation is unknown.

The t-distribution is used when one needs to estimate the population parameters (such as mean) but the population standard deviation is unknown. When $n$ is small (less than 30), one must be careful in invoking the normal distribution for $\overline{X}$. The distribution of $\overline{X}$ depends on the shape of the population distribution. Therefore, no single inferential procedure can be expected to work for all kinds of population distributions.

t Distribution

One Sample t-Test Formula

If $X_1, X_2, \cdots, X_n$ is a random sample from a normal population with mean $\mu$ and standard deviation of $\sigma$, the sample mean $\overline{X}$ is exactly distributed as normal with mean $\mu$ and standard deviation $\frac{\sigma}{\sqrt{n}}$ and $Z=\frac{\overline{X} – \mu}{\frac{\sigma}{\sqrt{n}}}$ is a standard normal variable. When $\sigma$ is unknown, the sample standard deviation is used, that is
$$t=\frac{\overline{X} – \mu}{\frac{s}{\sqrt{n}}},$$
which is analogous to the Z-statistic.

The Sampling Distribution for t

Consider samples of size $n$ drawn from a normal population with mean $\mu$ and for each sample, we compute $t$ using the sample $\overline{X}$ and sample standard deviation $S$ (or $s$), the sampling distribution for $t$ can be obtained

$$Y=\frac{k}{\left(1 + \frac{t^2}{n-1}\right)^{\frac{n}{2}} } = \frac{k}{\left(1+ \frac{t^2}{v} \right)^{\frac{v+1}{2} }},$$
where $k$ is a constant depending on $n$ such that the total area under the curve is one, and $v=n-1$ is called the number of degrees of freedom.

The t distributions are symmetric around zero but have thicker tails (more spread out) than the standard normal distribution. Note that with the large value of $n$, the t-distribution approaches the standard normal distribution.

Properties of the t Distribution

  • The t distribution is bell-shaped, unimodal, and symmetrical around the mean of zero (like the standard normal distribution)
  • The variance of the t-distribution is always greater than 1.
  • The shape of the t-distribution changes as the number of degrees of freedom changes. So, we have a family of $t$ distributions.
  • For small values of $n$, the distribution is considerably flatter around the center and more spread out than the normal distribution, but the t-distribution approaches the normal as the sample size increases without limit.
  • The mean and variance of the t distribution are $\mu=0$ and $\sigma^2 = \frac{v}{v-2}$, where $v>2$.

Common Application of t Distribution

  • t-tests are used to compared means between two groups
  • t-test are used to compared if a sample mean is significantly different from a hypothesized population mean.
  • t-values are used for constructing confidence intervals for population means when the population standard deviation is unknown.
  • Used to test the significance of the correlation and regression coefficients.
  • Used to construct confidence intervals of correlation and regression coefficients.
  • Used to estimate the standard error of various statistical models.

Assumptions of the t Distribution

The t-distribution relies on the following assumptions:

  • Independence: The observations in the sample must be independent of each other. This means that the value of one observation does not influence the value of another.
  • Normality: The population from which the sample is drawn should be normally distributed. However, the t-distribution is relatively robust to violations of this assumption, especially for larger sample sizes.
  • Homogeneity of Variance: If comparing two groups, the variances of the two populations should be equal. This assumption is important for accurate hypothesis testing.

Note that significant deviations from normality or unequal variances can affect the accuracy of the results. Therefore, it is always a good practice to check the assumptions before conducting a t-test and consider alternative non-parametric tests if the assumptions are not met.

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Download Student’s t Distribution Table

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MCQs Regression Analysis Quiz 7

The post is about the MCQs Regression Analysis Quiz with Answers. There are 20 multiple-choice questions from correlation analysis, regression analysis, correlation matrix, coefficient of determination, residuals, predicted values, Model selection, regularization techniques, etc. Let us start with the MCQs regression analysis quiz.

Online Multiple Choice Questions about Correlation and Regression Analysis

1. What is the difference between observed or actual values and the predicted values of a regression line?

 
 
 
 

2. Regression models are groups of ————– techniques that use data to estimate the relationships between a single dependent variable and one or more independent variables.

 
 
 
 

3. What variable selection process begins with the full model that has all possible independent variables?

 
 
 
 

4. What term describes an inverse relationship between two variables?

 
 
 
 

5. The best-fit line is the line that fits the data best by minimizing some —————.

 
 
 
 

6. What concept refers to how two independent variables affect the $Y$ dependent variable?

 
 
 
 

7. What does the circumflex symbol, or “hat” (^), indicate when used over a coefficient?

 
 
 
 

8. Which linear regression evaluation metric is sensitive to large errors?

 
 
 
 

9. Which of the following statements accurately describes a randomized, controlled experiment?

 
 
 
 

10. What type of visualization uses a series of scatterplots that show the relationships between pairs of variables?

 
 
 
 

11. Which of the following are regularized regression techniques?

 
 
 
 

12. ————- finds the mean of $Y$ given a particular value of $X$.

 
 
 
 

13. How does a data professional determine if a linearity assumption is met?

 
 
 
 

14. R squared measures the —————- in the dependent variable $Y$, which is explained by the independent variable, $X$.

 
 
 
 

15. ————- is a technique that estimates the relationship between a continuous dependent variable and one or more independent variables.

 
 
 
 

16. What is the sum of the squared differences between each observed value and the associated predicted value?

 
 
 
 

17. Which statements accurately describe coefficients and p-values for regression model interpretation?

 
 
 
 

18. Which of the following statements accurately describes the normality assumption?

 
 
 
 

19. Regression analysis aims to use math to define the ————– between the sample $X$’s and $Y$’s to understand how the variables interact.

 
 
 
 

20. Adjusted R squared is a variation of the R squared regression evaluation metric that ————— unnecessary explanatory variables.

 
 
 
 

MCQs Regression Analysis Quiz with Answers

MCQs Regression Analysis Quiz with Answers

  • What term describes an inverse relationship between two variables?
  • Regression analysis aims to use math to define the ————– between the sample $X$’s and $Y$’s to understand how the variables interact.
  • Regression models are groups of ————– techniques that use data to estimate the relationships between a single dependent variable and one or more independent variables.
  • ————- finds the mean of $Y$ given a particular value of $X$.
  • ————- is a technique that estimates the relationship between a continuous dependent variable and one or more independent variables.
  • The best-fit line is the line that fits the data best by minimizing some —————.
  • What is the sum of the squared differences between each observed value and the associated predicted value?
  • What does the circumflex symbol, or “hat” (^), indicate when used over a coefficient?
  • How does a data professional determine if a linearity assumption is met?
  • Which of the following statements accurately describes the normality assumption?
  • What type of visualization uses a series of scatterplots that show the relationships between pairs of variables?
  • R squared measures the —————- in the dependent variable $Y$, which is explained by the independent variable, $X$.
  • Which linear regression evaluation metric is sensitive to large errors?
  • Which statements accurately describe coefficients and p-values for regression model interpretation?
  • What is the difference between observed or actual values and the predicted values of a regression line?
  • Which of the following statements accurately describes a randomized, controlled experiment?
  • What concept refers to how two independent variables affect the $Y$ dependent variable?
  • Adjusted R squared is a variation of the R squared regression evaluation metric that ————— unnecessary explanatory variables.
  • What variable selection process begins with the full model that has all possible independent variables?
  • Which of the following are regularized regression techniques?
https://itfeature.com MCQS Regression Analysis Quiz with Answers

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Normal Probability Distribution

The Gaussian or normal probability distribution role is very important in statistics. It was investigated by researchers/ persons interested in gambling or in the distribution of errors made by people observing astronomical events. The normal probability distribution is important in other fields such as social sciences, behavioural statistics, business and management sciences, and engineering and technologies.

Importance of Normal Distribution

Some of the important reasons for the normal probability distribution are:

  • Many variables such as (weight, height, marks, measurement errors, IQ, etc.) are distributed as the symmetrical bell-shaped normal curve approximately.
  • Many inferential procedures (parametric tests: confidence intervals, hypothesis testing, regression analysis, etc.) assume that the variables follow the normal distribution.
  • All probability distributions approach a normal distribution under some conditions.
  • Even if a variable is not normally distributed, a distribution of sample sums or averages on that variable will be approximately normally distributed if the sample size is large enough.
  • The mathematics of a normal curve is well-known and relatively simple. One can find the probability that a score randomly sampled from a normal distribution falls within the interval $a$ and $b$ by integrating the normal probability density function (PDF) from $a$ to $b$. This is equivalent to finding the area under the curve between $a$ and $b$ assuming a total area of one.
  • Due to the Central Limit Theorem, the average of many independent random variables tends to follow a normal probability distribution, regardless of the original distribution of the variables.

Probability Density Functions of Normal Distribution

The probability density function known as the normal curve. $F(X)$ is the probability density, aka the height of the curve at value $X$.

$$F(X) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(X-\mu)^2}{2\sigma^2} }$$

There are two parameters in the PDF of normal distribution, (i) the mean and (ii) the standard deviation. Everything else in the PDF of normal distribution on the right-hand side is a constant. There is a family of normal probability distribution with respect to their means and their standard deviations.

Standard Normal Probability Distribution

One can work with normal curve, even if one don’t know about integral calculus. One can use computer to compute the area under the normal curve or make use of the normal curve table. The normal curve table (standard normal table) is based on the standard normal curve ($Z$), which has a mean of 0 and a variance of 1. To use a standard normal curve table, one need to convert the raw score to $Z$-scores. A $Z$-score is the number of standard deviations ($\sigma$ or $s$) a score is above or below the mean of a reference distribution.

$$Z_X = \frac{X-\mu}{\sigma}$$

For example, suppose one wish to know the percentile rank of a score of 90 on an IQ test with $\mu = 100$ and $\sigma=10$. The $Z$-score will be

$$Z=\frac{X-\mu}{\sigma} = \frac{90-100}{10} = -1$$

One can either integrate the normal cure from $-\infty$ to $-1$ or use the standard normal table. The probability or area under the curve on the left of $-1$ is 0.1587 or 15.87%.

Standard Normal Probability distribution Curve

Key Characteristics of Normal Probability Distribution

  • Symmetry: In normal probability distribution, the mean, median, and mode are all equal and located at the center of the curve.
  • Spread: In normal distribution, the spread of the data is determined by the standard deviation. A larger standard deviation means that the curve is wider, and a smaller standard deviation means a narrower curve.
  • Area under the Normal Curve: The total area under the normal curve is always equal to 1 or 100%.
normal curve dnorm() normal probability distribuiton

Real-Life Applications of Normal Distribution

The following are some real-life applications of normal probability distribution.

  • Natural Phenomena:
    • Biological Traits: Many biological traits, such as weight, height, and IQ scores, tend to follow a normal distribution. This helps us to understand the typical range of values for different biological traits and identify outliers.
    • Physical Measurements: Errors in measurements often follow a normal distribution. This knowledge is crucial in fields like engineering and physics for quality control and precision.
  • Statistical Inference:
    • Hypothesis Testing: The normal distribution is used extensively in hypothesis testing to determine the statistical significance of the results. By understanding the distribution of sample means, one can make inferences about population parameters.
    • Confidence Intervals: Normal distribution helps calculate confidence intervals, which provide a range of values within which a population parameter is likely to fall with a certain level of confidence.
  • Machine Learning and Artificial Intelligence:
    • Feature Distribution: Many machine learning (ML) algorithms assume that features in data follow a normal distribution. The normality assumption about machine learning algorithms can influence the choice of algorithms and the effectiveness of models.
    • Error Analysis: The normal distribution is used to analyze the distribution of errors in machine learning models, helping to identify potential biases and improve accuracy.
  • Finance and Economics:
    • Asset Returns: While not perfectly normal, many financial assets, such as stock prices, follow an approximately normal distribution over short time periods. The assumption of normality is used in various financial models and risk assessments.
    • Economic Indicators: Economic indicators such as GDP growth rates and inflation rates often exhibit a normal distribution, allowing economists to analyze trends and make predictions.
  • Quality Control:
    • Process Control Charts: In manufacturing and other industries, normal distribution is used to create control charts that monitor the quality of products or processes. By tracking the distribution of measurements, one can identify when a process is going out of control.
  • Product Quality: Manufacturers use statistical quality control methods based on normal distribution to ensure that products meet quality standards.
  • Everyday Life:
    • Standardized Tests: The standardized Test scores, such as SAT and GRE, are often normalized to a standard normal distribution, allowing for comparisons between different test-takers.
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