A Quick Overview of Probability

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The post is a quick overview of probability. Probability theory is a branch of mathematics that deals with the occurrence of random events. It provides a framework for quantifying uncertainty and making predictions based on available information.

Overview of Probability

The probability can be classified into two points of view:

Objective probability requires a computational formula, while subject probability can be derived from experience, judgment, or previous knowledge about the event. In this post, I will focus on an overview of probability and the Laws of Probability.

Objective Probability

The objective probability has the following definitions

Classical and a Priori Definition

$$P(A) = \frac{\text{Number of sample points in the event based on the favorable events}}{\text{Number of sample points in the sample space}} = \frac{m}{n} = \frac{n(A)}{n(S)}$$

The Relative Frequency or a Posteriori Definition

$$P(A) = \lim\limits_{n\rightarrow \infty} \frac{m}{n}$$

This definition assumes that as $n$ increases to infinity, $\frac{m}{n}$ becomes stable.

When we experiment with the same condition many times, the probability of favourable event becomes stable. For example, if we toss a coin 10 times, then 100 times, then 1000 times, then 10,000 times, then 100,000 times, then 1000,000, and so on. We are also interested in the various numbers of heads that occur. Let $H$ (occurrence of head) be our favorable event, and the probability of a favorable event is called the probability of success. Then the definition said that there are approximately 50% heads in one million tosses. This definition is also called the empirical or statistical definition of probability. that is more useful in practical problems. In practical problems, we find the winning percentage of a team.

The axiomatic Definition of Probability

An axiom is a statement, about any phenomenon, which is used to find real-world problems.

The axiomatic definition of probability states that if a sample space $S$ with sample points $E_1, E_2, \cdots, E_n$, then a real number is assigned to each sample point denoted by $P(E_i)$, should satisfy the following conditions:

  • for any event ($E_i$), $0< P(E_i) <1$
  • $P(S) = 1$, sure event
  • If $A$ and $B$ are two mutually exclusive events, then $P(A\cup B) = P(A) + P(B)$

Laws of Probability

For computing the probability of two or more events, the following laws of probability may be used.

Law of Addition

  • For mutually exclusive events: $P(A\cup B) = P(A) + P(B)$
  • For non-mutually exclusive events: $P(A\cup B) = P(A) + P(B) – P(A\cap B)$

If $A$, $B$, and $C$ are three events in a sample space $S$, then

$P(A\cup B \cup C) = P(A) + P(B) + P(C) – P(A \cap B) – P(B\cap C) – P(A \cap C)$

Law of Multiplication

For independent events $A$ and $B$: $P(A \text{ and } B) = P(A) \times P(B)$

For dependent events $A$ and $B$: $P(A \text { and } B) = P(A) \times P(B|A)$ (where $P(B|A)$ is the conditional probability of $B$ given $A$)

Law of Complementation

If $A$ is an event and $A’$ is the complement of that event, then

$P(A’) = 1-P(A)$, Note that $P(A) + P(A’) = 1$

Probability of sub-event

If $A$ and $B$ are two events in such a way that $A \subset B$, then $P(A) \le P(B)$

If $A$ and $B$ are any two events defined in a sample space $S$, then

$P(A\cap B’) = P(A) – P(A\cap B)$

Conditional Probability

$P(A|B) = \frac{P(A\cap B}{P(B)}$ or $P(B|A) = \frac{P(A\cap B}{P(A)}$.

Example of Conditional Probability

If we throw a die, what is the probability of 6? That is, $\frac{1}{6}$. What is the probability of 6 given that all are even numbers?

When a die is rolled, the sample space is $S=\{1, 2, 3, 4, 5, 6\}$. Let denote the even numbers by $B$, that is, $B=\{2, 4, 6\}$

$P(A|B) = \frac{1}{3}$

Law of Total Probability

If events $B_1, B_2, \cdots, B_n$ are mutually exclusive and exhaustive events, then for any event $A$: $P(A) = P(A|B_1) \times P(B_1) + P(A|B_2) * P(B_2) + \cdots + P(A|B_n) \times P(B_n)$

Bays’s theorem

Bays’ there is used to update probabilities based on new information.

If $A_1, A_2, \cdots, A_k$ are many events in a sample space.

$P(A_i|B) = \frac{P(A_i) P(B|A_i)}{\Sigma P(A)_i P(B|A_i)}, \text{ for } i, 1, 2, 3, \cdots, k$

Quick Overview of Probability

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Probability Distribution Discrete Random Variable

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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

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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.

statistics help: https://itfeature.com t distribution

Download Student’s t Distribution Table

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MCQs Regression Analysis Quiz 7

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

 
 
 
 

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

 
 
 
 

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

 
 
 
 

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

 
 
 
 

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

 
 
 
 

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

 
 
 
 

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

 
 
 
 

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

 
 
 
 

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

 
 
 
 

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

 
 
 
 

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

 
 
 
 

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

 
 
 
 

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

 
 
 
 

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

 
 
 
 

15. Which of the following are regularized regression techniques?

 
 
 
 

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

 
 
 
 

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

 
 
 
 

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

 
 
 
 

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

 
 
 
 

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?
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