Statistics for Doctors #3: Probability Distribution

Statistics for Doctors: Probability Distributions Having discussed the types of probability now we can talk about how it presents. We typically call this probability distribution. A probability distribution describes the likelihood of different outcomes in a random experiment. It quantifies the uncertainty associated with the possible values that a random variable can take. It is usually denoted as P[X = x], where X is the random variable and x is the values it can take. A random variable is a variable whose values are determined by the outcome of a random experiment. There are two main types of probability distributions, discrete and continuous. Discrete Distribution A discrete distribution is a probability model that shows the likelihood of different separate outcomes happening in a random experiment. It involves a random variable with finite or countably infinite distinct values, each with its probability of occurring. In simpler terms, a discrete distribution describes the probabilities associated with specific, separate outcomes, where each outcome has a well-defined chance of occurring. We would need discrete distribution to model scenarios where outcomes are distinct and countable. Strengths would be that it provides a straightforward representation of the probability of each outcome, making it easy to interpret, and it is well suited for scenarios where outcomes are distinct and countable and it is easier to work, allowing simpler calculations. Weaknesses would be that it would not adequately capture the variability present in real-world phenomena, and like all probability distributions it is based on certain assumptions on the underlying processes and their accuracy depends on the validity of these assumptions, which may not always be true in practice. For example, mammal X is known for having variable sleep durations. The probability of a mammal sleeping for 8 h per day is 0.4, meaning 40% of observed instances involve mammals sleeping for 8 h. Continuous Distribution This type represents the likelihood of various continuous outcomes occurring in a random experiment. In a continuous distribution, the random variable can take on any value within a certain range, and the probability of any specific value occurring is typically zero. Instead, probabilities are associated with intervals or ranges of values. In simpler terms, a continuous distribution describes the probabilities associated with a continuous range of outcomes, where each possible value within the range contributes to the overall probability distribution. They are essential for scenarios where outcomes are continuous and can take on any value within a range. Strengths would be that it is suitable for continuous outcomes, such as height, weight, temperature, and time intervals, and provides precise probabilities for events with continuous range, enabling detailed analysis and prediction of outcomes. Weaknesses would be that it often involves complex mathematical formulas, which can be intensive and require mathematical knowledge to work effectively, interpreting these may be less intuitive, and like all probability distributions, they are based on assumptions, which may not always be true in practice. Examples of such continuous distributions are normal and exponential distributions. Example—Imagine mammal X usually sleeps for an average of 10 h each day. We can use the exponential distribution to understand how likely it is for mammal X to wake up at any given moment during its sleep. Binomial distribution The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent and identical trials. Each trial has two possible outcomes: success and failure. The probability of success on each trial is assumed to be constant. We need this to model scenarios where there are only two possible outcomes for which trail, and the trails are independent and have the same probability. Strengths would be that it is straightforward and accessible even to those with limited statistical knowledge and it allows for precise calculation of probabilities of obtaining a specific number in a fixed number of trials It forms a basis for hypothesis testing and confidence interval estimation. Weaknesses would be that it is not suitable for scenarios with more than two outcomes. It assumes each trial is independent, which may not always be true and it is not appropriate for a small sample size. Example—Consider the quantitative variable order (primates vs. nonprimates). Each mammal in the dataset has a certain chance of being a primate and therefore the total number of primates in the data set will be binomially distributed. This is shown in Table 1.Table 1: Binomial distribution of primatesNormal distribution The normal distribution is a continuous probability distribution that is, symmetric around its mean. It is characterized by mean and standard deviation. About 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. The normal curve can be used for the graphical representation of the normal distribution. It is a smooth, symmetric, bell-shaped curve that describes the probability density function of the normal distribution. The area under the normal curve represents the probability of observing values within a certain range. Strengths would be that it is widely applicable, and serves as a fundamental concept in statistics, as the distribution of sample means from any population approaches a normal distribution, a bell-shaped curve makes it easy to interpret, and it is fully described as mean and standard deviation. Weaknesses would be that it may not represent all real-world data, especially those with skewness, and it extends infinitely in both directions, which can lead to impracticalities when dealing with bounded data. The sample mean of random samples taken from the total sleep duration of mammals is normally distributed. This is shown in Figure 1.Figure 1: Normal distribution of sample mean of samples taken from total sleep time of mammals. The y-axis shows the probability density, and the x-axis shows the sample means of 1000 samples taken from the total sleep duration. The normal curve overlaid on the histogram represents the normal distribution that closely fits the observed data. The mean of the distribution is 10.46 and the standard deviation is 0.81Sampling distribution Sampling distribution describes the likelihood of obtaining a particular sample statistic based on a random sampling process. It is a theoretical distribution that represents all possible values of a sample statistic such as mean and standard deviation, when repeated samples are drawn from a population. The sampling distribution of sample mean describes the distribution of sample means obtained from repeated random samples of a fixed size taken from a population. The sampling distribution of sleep duration provides information about the variability and precision of sample means obtained from different samples drawn from the population of mammals. Strengths would be that it allows one to make estimates about population parameters, construct hypothesis tests, and construct confidence intervals based on sample data, they are applicable across various statistical methods and analysis, making them fundamental concepts in statistical theory and practice. Weaknesses would be that conducting the analysis can be computationally intensive, especially for large data sets. The schematic representation of this sampling distribution is shown in Figure 2.Figure 2: Schematic representation of sampling distribution of sleep duration. We drew 1000 random samples, each of size 30, from the total sleep duration of mammals with mean µ. For each sample, we computed the mean sleep duration ȳ, resulting in a sampling distribution of 1000 sample meansFinancial support and sponsorship Nil. Conflicts of interest There are no conflicts of interest.