probability mass function

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 $p$.

• Probability

  • Probability density function describes the relative likelihood, or probability, that a given variable will take on a value.
  • Here, we will learn what probability distribution function is and how it functions with regard to integration.
  • In probability theory, a probability density function (pdf), or density of a continuous random variable, is a function that describes the relative likelihood for this random variable to take on a given value.
  • A probability density function is most commonly associated with absolutely continuous univariate distributions.
  • Apply the ideas of integration to probability functions used in statistics

• Quantum Tunneling

  • Believe it or not, there is a finite (if extremely small) probability that this event would occur.
  • The answer is a combination of the tunneling object's mass ($m$) and energy ($E$) and the energy height ($U_0$) of the barrier through which it must travel to get to the other side.
  • When it reaches a barrier it cannot overcome, a particle's wave function changes from sinusoidal to exponentially diminishing in form.
  • Therefore, the probability of an object tunneling through a barrier decreases with the object's increasing mass and with the increasing gap between the energy of the object and the energy of the barrier.
  • And although the wave function never quite reaches 0 (as can be determined from the $e^{-x}$ functionality), this explains how tunneling is frequent on nanoscale but negligible at the macroscopic level.

• 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