Discrete and Continuous Probability Distributions: PMF, CDF, Rayleigh, Binomial, and Poisson

Published on October 4, 2026

Statistics

Solve a discrete random-variable problem by finding the normalization constant, probability mass function, and cumulative distribution function. The answer also explains continuous probability density functions, the Rayleigh distribution, the binomial pmf, and the Poisson probability function.

Question

Give a 10m answers

Answer

1. Discrete random variable: find , and

Since the probabilities must sum to

Thus the probability mass function is

The cumulative distribution function

To draw the plots, plot the probability masses at the six listed values of

. The CDF is a right-continuous step function, with jumps of sizes

, respectively.

2. Probability density function of a continuous random variable

A continuous random variable

has probability density function (pdf)

satisfying:

for all .
The total area under the density curve is :
Probabilities over intervals are areas under the curve:
The CDF isConsequently, where the derivative exists.
For any single value , .

A density curve is nonnegative, and the entire area beneath it equals

 fX(x)
   ^
   |            /\
   |          /    \
   |________/________\________> x
              total area = 1

3. Rayleigh density function

For a parameter

, the Rayleigh pdf is

It is a valid density since it is nonnegative and

Its CDF is

The density starts at

, reaches its maximum at

, and then decreases toward

. The Rayleigh distribution occurs, for example, as the magnitude of two independent zero-mean normal random variables with the same variance

 fX(x)
   ^
   |       /\
   |      /  \____
   |_____/___________> x
        0  σ

4. Binomial probability mass function and Poisson distribution

Binomial distribution

counts the number of successes in

independent trials, each with success probability

, then

has a binomial distribution:

Here

. The probabilities sum to

, and

Its graph consists of probability bars at the integers

Poisson probability mass function

A Poisson random variable with parameter

has pmf

with

For example, when

, the probabilities are plotted as discrete bars:

P(X=x)
  ^
  |          █
  |       █  █  █
  |    █  █  █  █  █
  | █  █  █  █  █  █  █
  +----------------------> x
    0  1  2  3  4  5  6

The Poisson graph is discrete; its bars are centered at nonnegative integers. Its shape depends on

, with the mass concentrated around

Theorems

• Total Probability of a Discrete Random Variable

The probabilities of all possible values of a discrete random variable sum to

• Probability Mass Function (PMF)

For a discrete random variable, the PMF gives the probability of each value:

It satisfies

and

• Cumulative Distribution Function (CDF)

The CDF gives the probability that a random variable is at most

For a discrete variable,

• Probability Density Function (PDF)

A continuous random variable has a nonnegative density

with total integral

Interval probabilities are areas under the density:

• CDF–PDF Relationship

For a continuous random variable, the CDF is the integral of its density:

Where differentiable,

• Binomial Probability Formula

counts successes in

independent trials, each having success probability

, then

Its mean and variance are

and

• Poisson Probability Formula

A Poisson random variable with parameter

has probabilities

Its mean and variance are both

• Rayleigh Distribution

For

, the Rayleigh density is

and

for

. Its CDF for

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