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A probability distribution is a statistical function that describes all possible values of a random variable within a given range and the likelihood of their occurrence. The range is bounded by a minimum and maximum value.
A probability distribution is a statistical function that describes all possible values of a random variable within a given range and the likelihood of their occurrence. The range is bounded by a minimum and maximum value. However, the exact location of the possible values on the probability distribution depends on several factors, including the mean (mean), standard deviation, skewness, and kurtosis of the distribution.
Perhaps the most common probability distribution is the normal distribution, or bell curve, although a variety of other distributions are widely used. The data generation process for some phenomenon often determines its probability distribution. This process is called the probability density function.
Probability distributions can also be used to generate a cumulative distribution function (CDF), which adds up the probability of an event occurring. They always start from zero and end up at 100%. [1]
Academics, financial analysts, and fund managers may evaluate probability distributions for a particular stock to assess the expected returns that stock may deliver in the future.
IMPORTANT NOTE: Historical data on stock returns can be measured from any time interval and may only include a portion of stock returns. This makes the analysis subject to sampling error. This error can be significantly reduced by increasing the sample size.
Discrete probability distributions and continuous probability distributions are two basic types of probability distributions, each describing different types of random variables. Understanding their differences is critical to correctly applying statistical methods and interpreting data.
Discrete probability distributions describe situations in which the set of possible outcomes is countable and finite or countably infinite. These distributions are used when a random variable can take on specific, clear values.
For example, the number of heads in 10 coin tosses or the number of customers arriving at a store in an hour are examples of discrete random variables. In these scenarios, you can list all possible outcomes, such as zero, one, two, and so on. Outcomes from a discrete probability distribution are more likely to "jump" because there are fewer outcomes.
In contrast, a continuous probability distribution is suitable for situations where the random variable can take any value within a given range. These values are uncountable because there are countless possibilities within any interval.
For example, the exact height of an individual in a population or the exact time it takes to complete a task are continuous variables. The distribution curve of a continuous probability distribution may appear smoother because there are more possible outcomes.
There are many classifications of probability distributions. They include the normal distribution, chi-square distribution, binomial distribution, and Poisson distribution. These probability distributions serve different purposes and represent different data-generating processes. [1][2]
The binomial distribution evaluates the probability that an event will occur more than once in a given number of trials, with this probability being fixed in each trial. It can be generated by recording the number of free throws a basketball player makes during a game, where 1 indicates a goal and 0 indicates a missed goal.
Another example is using a coin and calculating the probability that the coin will come up heads in 10 consecutive tosses. The binomial distribution is discrete because the valid outcomes can only be 0 or 1. [3]
The most commonly used distribution is the normal distribution. It is frequently used in fields such as finance, investment, science, and engineering. A normal distribution is fully characterized by its mean and standard deviation. The distribution is not skewed and exhibits kurtosis.
This gives the distribution a symmetrical shape, which appears as a bell-shaped curve when plotted. A normal distribution has a mean (mean) of zero, a standard deviation of one, a skewness of zero, and kurtosis = 3.
In a normal distribution, about 68% of the data will fall within plus or minus one standard deviation of the mean. About 95% of the data will fall within plus or minus two standard deviations of the mean, and 99.7% of the data will fall within plus or minus three standard deviations of the mean. Unlike the binomial distribution, the normal distribution is continuous. All possible values are represented, not just no other values between 0 and 1. [4]
Note: Probability is the likelihood of an event occurring and is the ratio of a favorable outcome to the total number of all possible outcomes. Odds, on the other hand, express the ratio of the probability of an event occurring to the probability of not occurring. For example, if the probability of winning the game is 0.25, the odds are 1:3 (one win for three losses).
The law of large numbers states that as the number of trials increases, the average of the results obtained will approach the expected value or true probability. This principle ensures that as the number of observations increases, the sample mean will converge to the population mean, thus providing stability for statistical inference.
Determining whether a probability distribution is valid requires two steps. The first step is to analyze whether each probability is greater than or equal to zero and less than or equal to one. The second step is to determine whether the sum of all probabilities is equal to one. This probability distribution is valid if both the first and second steps are true.
Probability distributions have two main applications in the financial field:
The most commonly used probability distributions include the uniform, binomial, Bernoulli, normal, Poisson, and exponential distributions. [6]
Probability measures the likelihood of an event occurring, expressed as the ratio of the number of favorable outcomes to the number of all possible outcomes. Odds represent the ratio of the probability of an event occurring to the probability of it not occurring. For example, if the probability of winning the game is 0.25, the odds are 1:3 (one win for three losses).
The Central Limit Theorem (CLT) is a statistical principle that states that the distribution of the sum of a large number of independent, identically distributed random variables approaches a normal distribution.
The importance of this theorem is that it allows statisticians to make inferences about population parameters even when the population distribution is unknown, as long as the sample size is large enough.
A key implication of CLT is that when the sample size is large, the sampling distribution of the sample mean will approximately follow a normal distribution.
For example, imagine you have a class where students vary in height, but the average height is about 5 feet, with some variation. According to CLT, the distribution of average height samples will tend to be a normal (bell-shaped) curve.
Observe what happens when you throw two standard six-sided dice. The probability of each die rolling any number (1 to 6) is 1/6, but the sum of the two dice forms a probability distribution as shown. 7 is the most common result (1+6, 6+1, 5+2, 2+5, 3+4, 4+3). 2 and 12 appear much less frequently (1+1 and 6+6).
A probability distribution describes all the values a random variable can take. This is widely used in investing, particularly in assessing a stock's potential performance and in risk management to help determine maximum losses.
[1] PennState, Eberly College. "STAT 500 Applied Statistics; Lesson 3: Probability Distributions."
[2] Dartmouth Department of Mathematics. "Grinstead and Snell’s Introduction to Probability." Pages 233-234.
[3] PennState, Eberly College. "STAT 500 Applied Statistics; 3.2.2 - Binomial Random Variables."
[4] University of Pennsylvania. "2.2.7 - The Empirical Rule."
[5] U.S. Securities and Exchange Commission. “Remarks Before the Peterson Institute of International Economics.”
[6] Dartmouth Department of Mathematics. "Grinstead and Snell’s Introduction to Probability."
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