probability mass function
(noun)
a function that gives the probability that a discrete random variable is exactly equal to some value
Examples of probability mass function in the following topics:
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Probability Distributions for Discrete Random Variables
- This can be expressed through the function $f(x)= \frac{x}{10}$, $x=2, 3, 5$ or through the table below.
- Sometimes, the discrete probability distribution is referred to as the probability mass function (pmf).
- The probability mass function has the same purpose as the probability histogram, and displays specific probabilities for each discrete random variable.
- This shows the graph of a probability mass function.
- All the values of this function must be non-negative and sum up to 1.
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Two Types of Random Variables
- The mathematical function describing the possible values of a random variable and their associated probabilities is known as a probability distribution.
- Their probability distribution is given by a probability mass function which directly maps each value of the random variable to a probability.
- Continuous random variables, on the other hand, take on values that vary continuously within one or more real intervals, and have a cumulative distribution function (CDF) that is absolutely continuous.
- The image shows the probability density function (pdf) of the normal distribution, also called Gaussian or "bell curve", the most important continuous random distribution.
- This shows the probability mass function of a discrete probability distribution.
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The Hypergeometric Random Variable
- The hypergeometric distribution is a discrete probability distribution that describes the probability of $k$ successes in $n$ draws without replacement from a finite population of size $N$ containing a maximum of $K$ successes.
- A random variable follows the hypergeometric distribution if its probability mass function is given by:
- In the softball example, the probability of picking a women first is $\frac{13}{24}$.
- The probability of picking a man second is $\frac{11}{23}$, if a woman was picked first.
- The probability of the second pick depends on what happened in the first pick.
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The Binomial Formula
- Table 1 is a discrete probability distribution: It shows the probability for each of the values on the $x$-axis.
- In probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of $n$ independent yes/no experiments, each of which yields success with probability $p$.
- The probability of getting exactly $k$ successes in $n$ trials is given by the Probability Mass Function:
- When $n$ is relatively large (say at least 30), the Central Limit Theorem implies that the binomial distribution is well-approximated by the corresponding normal density function with parameters $\mu = np$ and $\sigma = \sqrt{npq}$.
- Employ the probability mass function to determine the probability of success in a given amount of trials
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Binomial Probability Distributions
- This chapter explores Bernoulli experiments and the probability distributions of binomial random variables.
- It is a discrete probability distribution with two parameters, traditionally indicated by $n$, the number of trials, and $p$, the probability of success.
- Since the trials are independent and since the probabilities of success and failure on each trial are, respectively, $p$ and $q=1-p$, the probability of each of these ways is $p^x(1-p)^{n-x}$.
- These probabilities are called binomial probabilities, and the random variable $X$ is said to have a binomial distribution.
- A graph of binomial probability distributions that vary according to their corresponding values for $n$ and $p$.
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The Poisson Random Variable
- What is the probability that exactly 3 homes will be sold tomorrow?
- Thus, the probability of selling 3 homes tomorrow is 0.180.
- The Poisson distribution is a discrete probability distribution.
- What is the probability that exactly 3 homes will be sold tomorrow?
- The function is only defined at integer values of k.
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Continuous Probability Distributions
- A continuous probability distribution is a probability distribution that has a probability density function.
- In theory, a probability density function is a function that describes the relative likelihood for a random variable to take on a given value.
- Unlike a probability, a probability density function can take on values greater than one.
- The standard normal distribution has probability density function:
- Boxplot and probability density function of a normal distribution $$$N(0, 2)$.
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Common Discrete Probability Distribution Functions
- Some of the more common discrete probability functions are binomial, geometric, hypergeometric, and Poisson.
- A probability distribution function is a pattern.
- You try to fit a probability problem into a pattern or distribution in order to perform the necessary calculations.
- These distributions are tools to make solving probability problems easier.
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Student Learning Outcomes
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Properties of Continuous Probability Distributions
- The curve is called the probability density function (abbreviated: pdf).
- We use the symbol f (x) to represent the curve. f (x) is the function that corresponds to the graph; we use the density function f (x) to draw the graph of the probability distribution.
- Area under the curve is given by a different function called the cumulative distribution function (abbreviated: cdf).
- The cumulative distribution function is used to evaluate probability as area.
- In general, calculus is needed to find the area under the curve for many probability density functions.