probability mass function

(noun)

a function that gives the relative probability that a discrete random variable is exactly equal to some value

Related Terms

  • probability distribution
  • central limit theorem
  • discrete random variable

(noun)

a function that gives the probability that a discrete random variable is exactly equal to some value

Related Terms

  • probability distribution
  • central limit theorem
  • discrete random variable

Examples of probability mass function in the following topics:

  • 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.
  • 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.
  • 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.
  • 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
  • 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 $p$.
  • 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.
  • 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)$.
  • 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.
  • Student Learning Outcomes

  • 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.
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