The Poisson distribution, the spread of the distribution around its mean is equal to the mean itself.

If we consider the Poisson distribution with λ = 0.5, the plot would show that the probability of observing zero events is the highest, followed by the probability of observing one event, and so on. As λ increases to 1 and 2, the blot shifts to the right, indicating that the probability of observing more events increases. However, this does not imply that larger λ means we have more trials. The Poisson distribution describes a scenario where events occur randomly in time or space, and the value of λ represents the average rate of events per unit of time or space.

For the Poisson distribution it is always the case that the variance is equal to the expectation, namely to λ: E(X) = Var(X) = λ is that because when the Var(X) = np(1-p) and p is small, (1-p) is approaching 1 , correct?

It is actually No, the statement E(X) = Var(X) = λ for the Poisson distribution is actually true regardless of the value of p, as the Poisson distribution is not related to the binomial distribution formula of Var(X) = np(1-p).

The Poisson distribution has a single parameter λ which represents both its mean and variance. Specifically, the mean and variance of a Poisson random variable X are both equal to λ, i.e., E(X) = Var(X) = λ. This is a fundamental property of the Poisson distribution that can be derived mathematically from its probability mass function.

The equality E(X) = Var(X) = λ for the Poisson distribution means that the spread of the distribution around its mean is equal to the mean itself. In other words, the Poisson distribution has a specific "shape" where the probability of observing values around the mean is highest, and this probability decreases as we move further away from the mean. This is often visualized as a bell curve or a symmetric "hump" centered around the mean value.

When E(X) = Var(X) = λ, it also means that the probability of observing very large or very small values is relatively low. For example, if λ = 5, the probability of observing a value of 10 or more is only about 0.02, and the probability of observing a value of 20 or more is less than 0.0001. On the other hand, the probability of observing a value of 4 or less is about 0.26, and the probability of observing a value of 3 or less is about 0.12.

In summary, E(X) = Var(X) = λ for the Poisson distribution tells us about the expected value and the spread of the distribution, as well as the probabilities of observing different values. This property is useful in many applications, such as modeling rare events, counting occurrences of certain phenomena, and analyzing queuing systems.

Reference

Yakir, B. (2011). Introduction to statistical thinking (with R, without Calculus). The Hebrew University of Jerusalem, Department of Statistics.

How either the Poisson or the Exponential distribution could be used to model something in real life?

When dealing with two types of discrete random variables, the Binomial and the Poisson, and two types of continuous random variables, the Uniform and the Exponential. Depending on the context, these types of random variables may serve as theoretical models of the uncertainty associated with the outcome of a measurement.

One example of how the Poisson distribution could be used to model something in real life is to estimate the number of calls a call center may receive during a certain period. The sample space in this case would be the set of all possible numbers of calls that the call center may receive during a specific time interval, such as one hour or one day.

The Poisson distribution is a discrete probability distribution that models the number of events occurring in a fixed interval of time or space, given that the events occur independently and at a constant rate. Thus, it can be used to estimate the probability of a certain number of calls during a certain period, given the historical data on the average rate of calls. The Exponential distribution, on the other hand, could be used to model the time between successive calls in a call center, assuming that calls arrive according to a Poisson process. The sample space in this case would be the set of all possible time intervals between successive calls. The Exponential distribution is a continuous probability distribution that models the time between events occurring independently and at a constant rate. Thus, it can be used to estimate the probability of a certain time interval between two successive calls, given the historical data on the average rate of calls. Moreover, another example of how the Poisson distribution could be used to model something in real life is to model the number of earthquakes that occur in a certain region over a given period of time. In this case, the sample space would consist of all possible counts of earthquakes that could occur in that region within the specified time frame.

Having a theoretical model for a situation can be important in many ways. It can help to predict future events, optimize processes, and make informed decisions. For example, in the case of a call center, knowing the probability distribution of the number of calls and the time between calls can help to optimize the staffing levels, allocate resources efficiently, and provide better customer service. However, it is important to note that theoretical models are simplifications of reality and may not always perfectly capture all the relevant factors. Therefore, they should be used in conjunction with empirical data and expert knowledge.

To summarize, the Poisson distribution is often used in situations where we are interested in the number of events that occur in a fixed interval of time or space such as to model the number of customers arriving at a store during a specific hour, the number of accidents occurring on a certain road during a day, or the number of calls received by a call center during a certain period of time. Having a theoretical model for the situation is important because it allows us to make predictions about the future based on historical data. For example, if we know the historical frequency of earthquakes in a region, we can use the Poisson distribution to predict the likelihood of earthquakes occurring in the future. In general, having a theoretical model allows us to understand the underlying structure of the data and make predictions based on that understanding. Without a theoretical model, we may be forced to rely on purely empirical methods, which may not be as accurate or effective in predicting future outcomes.

Reference

Yakir, B. (2011). Introduction to statistical thinking (with R, without Calculus). The Hebrew University of Jerusalem, Department of Statistics.

Binomial Distribution, the number of successes in n independent and identically distributed trials, rather than a specific value of the variable being observed.

For students who are confused by the concept of Binomial Distribution. At first, I was thinking that if X ∼ Binomial(10, 0.5), where n = 10 is the number of trials and p = 0.5 is the probability of success in each trial. What is the code x <- 0:10 and then dbinom(x,10,0.5) in the R means? Why the expection E(X) is not simply each value times its probability like we did previously?

The code x<-0:10 creates a vector of integers ranging from 0 to 10, inclusive. This vector represents the possible values of the number of successes, denoted by x, in a binomial distribution with parameters n=10 and p=0.5. The function dbinom(x,10,0.5) computes the probability mass function of the binomial distribution with parameters n=10 and p=0.5 at each value in the vector x. The output is a vector of probabilities, where each element represents the probability of observing the corresponding value of x in the binomial distribution.

For example, suppose we want to calculate the probability of getting exactly 5 heads in 10 coin tosses, where the coin is fair. We can use the binomial distribution with n=10 and p=0.5 to model this situation. First, we create a vector of possible values for the number of heads, from 0 to 10, using the x<-0:10 command. Then, we can use the dbinom() function to calculate the probability of getting exactly 5 heads, given the binomial distribution:

x <- 0:10

dbinom(5, 10, 0.5)

The output should be 0.2460938, which represents the probability of getting exactly 5 heads in 10 coin tosses, where the probability of getting heads is 0.5.

The most important concept which confused me here is that in the context of a binomial distribution, the value of x represents the number of successes in n independent and identically distributed trials, each with a probability of success p, rather than a specific value of the variable being observed.

Relative Frequency and Probability, What's The Difference?

 Relative frequency and probability are related concepts, but they are not exactly the same. Relative frequency refers to the proportion or fraction of times that an event occurs in a given set of data. It is calculated by dividing the frequency of the event by the total number of observations in the data set. For example, if we observe 50 heads in 100 coin tosses, the relative frequency of getting heads is 50/100 = 0.5 or 50%. Probability, on the other hand, refers to the likelihood or chance of an event occurring. It is a measure of how likely or unlikely an event is, and it is usually expressed as a number between 0 and 1 (or between 0% and 100%). Probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. For example, the probability of getting heads on a fair coin is 0.5 or 50%.

The difference between relative frequency and probability is that relative frequency is based on observed data, while probability is based on a theoretical or assumed model of the underlying process. Probability is a mathematical concept that allows us to reason about the likelihood of events, even when we don't have access to the data or when the data is incomplete. Relative frequency, on the other hand, is a tool for analyzing the distribution of data and making inferences based on the observed patterns.

It is also incorrect for finding the expectation of a random variable from a relative frequency table. To find the expectation of a random variable, we need to multiply each value of the random variable by its corresponding probability and sum up the products. In a relative frequency table, the probabilities are given by the relative frequencies, which are obtained by dividing the frequency of each value by the total number of observations. Therefore, to find the expectation of a random variable from a relative frequency table, we need to first convert the table into a probability table by dividing the frequencies by the total number of observations.

The difference between x̄ (x-bar) and μ (mu)

 In statistics, the symbol x̄ (x-bar) represents the sample mean or average of a set of data. It is calculated by adding up all of the values in the sample and dividing by the total number of values in the sample. The Greek letter mu (μ) represents the population mean or average of a larger group of data that the sample is drawn from. It is calculated in the same way as x-bar, but it represents the true mean of the entire population, rather than just the sample.

The difference between x̄ (x-bar) and μ (mu) is that x-bar represents the average of a sample of data, while mu represents the true average of the entire population from which the sample is drawn. Because samples are inherently imperfect and may not perfectly reflect the larger population, x-bar and mu may be different from each other.

Reference
Yakir, B. (2011). Introduction to statistical thinking (with R, without Calculus). The Hebrew University of Jerusalem, Department of Statistics.

The Reasons Why Measurements May Not Be Perfectly Reproducible

About the reproducible of a sample, I reckon that there are several reasons why measurements may not be perfectly reproducible, even when the same phenomenon is being measured under apparently identical conditions:

i. Measurement errors

All measuring instruments have some degree of imprecision or error associated with them. For example, a ruler may not be exactly straight, or a thermometer may not be calibrated perfectly. These errors can accumulate over repeated measurements and contribute to variability in the outcomes.

ii. Environmental factors

Even seemingly small differences in the environment can affect measurements. For example, changes in temperature, humidity, or air pressure can influence the behavior of some measuring instruments.

iii. Human factors

The people conducting the measurements may introduce variability due to their own limitations. For example, they may have slightly different visual acuity or reaction times, or they may interpret the results differently.

iv. Inherent variability

Some phenomena are inherently variable, and measurements of them will naturally vary. For example, in biology, there may be natural variation in the characteristics of organisms, even within a single population.

v. Random chance: Finally, there is always an element of chance involved in any measurement. Even if all sources of variability were eliminated, there would still be some residual randomness that would make it impossible to achieve perfectly reproducible results.

Overall, it is important to recognize that variability in measurements is a natural and unavoidable aspect of scientific research. However, scientists use statistical methods to quantify and manage this variability, in order to draw reliable conclusions from their data.

Random Variables Are Used To Model Situations In Which The Outcome, Before The Fact, Is Uncertain

For this week’s discussion, let's consider an example of measuring the effectiveness of a marketing campaign using a random variable. The variable we might choose to measure could be the number of website visits generated by the campaign. The sample space for this variable would be the range of possible website visits, which could be any positive integer value. The probabilities associated with each value in the sample space would depend on the success of the marketing campaign. If the campaign is successful, we might expect higher probabilities for values near the upper end of the sample space, while a less effective campaign might have more uniform probabilities across the range of possible values.

Suppose the goal of the campaign is to generate website visits, and the marketing team has developed a plan to drive traffic to the website through a combination of social media advertising, email marketing, and search engine optimization. The team has set a target of 10,000 website visits over a one-month period for the campaign. However, due to the unpredictable nature of marketing, the actual number of website visits generated by the campaign may vary from this target.

To model this situation using a random variable, we could define the variable X as the number of website visits generated by the campaign over the one-month period. The sample space for X would be the set of all possible positive integer values that X could take on, from 0 to some upper limit. Let's say that we define the upper limit as 20,000 website visits. We can then assign probabilities to each value in the sample space based on our expectations for the success of the campaign. For instance, we might estimate that the probability of generating 10,000 website visits (our target) is 0.3, while the probabilities of generating fewer or more visits could be distributed according to a normal distribution with a mean of 10,000 and a standard deviation of 2,500. Using these probabilities, we can make statistical predictions about the likely range of website visits, and calculate the expected value and variance of the random variable X.

However, it's important to remember that any individual outcome is subject to random variation and unpredictable factors, even if the overall success of the campaign is well-modeled by the random variable X. For example, a major news event or a competitor's marketing campaign could impact consumer behavior in unexpected ways and result in more or fewer website visits than predicted. As a result, while random variables can be useful for modeling marketing effectiveness.

As mentioned above, a random variable is the future outcome of a measurement, before the measurement is taken. If someone claims to know the outcome of an individual observation of website visits generated by the marketing campaign, they are likely overestimating their ability to predict the outcome. While we can use the sample space and associated probabilities to make statistical predictions about the likely range of website visits, any individual outcome will still be subject to random variation. Factors outside of the campaign, such as seasonality, external events, or changes in consumer behavior, could impact the outcome in ways that are difficult to predict.

To summarize, random variables can be useful for modeling situations in which the outcome is uncertain due to random variation, such as marketing effectiveness. While the sample space and associated probabilities can help to provide insights into the range of possible outcomes, it is important to acknowledge that any single outcome is still unpredictable and subject to random variation.

Reference
Yakir, B. (2011). Introduction to statistical thinking (with R, without Calculus). The Hebrew University of Jerusalem, Department of Statistics.