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Probability Distributions and Expected Values of Discrete Random Variables, Study notes of Mathematics

Mathematics

This document illustrates the concept of random variables, both discrete and continuous, and their probability distributions. It also discusses the construction of probability mass functions and the calculation of probabilities for discrete random variables. The document then provides examples of calculating the expected values of discrete random variables.

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2019/2020

Uploaded on 04/15/2020

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illustrate a random variable (discrete and

continuous)

distinguish between a discrete and a

continuous random variable

find possible values of a random variable

illustrate probability distributions for discrete

random variable and its properties

Partial preview of the text

Download Probability Distributions and Expected Values of Discrete Random Variables and more Study notes Mathematics in PDF only on Docsity!

illustrate a random variable (discrete and

continuous)

distinguish between a discrete and a

continuous random variable

find possible values of a random variable

illustrate probability distributions for discrete

random variable and its properties

An experiment consists of rolling a pair of dice.

Let Y be the random variable representing the

absolute value of the difference of the numbers

that come up. Construct the probability

distribution of Y in tabular, graphical and

formula form.

the values of random variables for which the

probability is greater than zero: 𝑃 𝑥 > 0

*the mass points are: 0, 1, 2, 3, 4 and 5

Y 0 1 2 3 4 5 P(y) 1 6

5 18

2 9

1 6

1 9

1 18

The possible values (Support S) of a random

variable is the set of values that a random

variable can take. This means that the elements

of S can be put into one-to-one correspondence

with the set of natural numbers.

 Let 𝐹 𝑥 = 𝑘𝑥 for 𝑥 = 1, 2, 3, 4. Find 𝑘 so that 𝐹(𝑥) satisfies the two properties of being a probability mass function. 𝐹 𝑥 𝑥∈𝑆

= 𝑘𝑥

𝑥=

= 1 1 = 𝑘 1 + 𝑘 2 + 𝑘 3 + 𝑘 4 1 = 𝑘 + 2𝑘 + 3𝑘 + 4𝑘 1 = 10𝑘

1 10

by substitution:

𝐹 𝑥 = 1 =

10 𝐹 𝑥 = 3^ =

10 𝐹 𝑥 = 4^ =

*therefore 0 ≤ 𝐹 𝑥 ≤ 1 (first property satisfied)

𝐹 𝑥 =

*therefore the sum of the probabilities is equal to 1 (second property satisfied)

Consider the formula:

𝑃 𝑥 =

Show that 𝑃 𝑥 is indeed a pmf.

Consider the formula:

𝑃 𝑥 =

Show that P 𝑥 is indeed a pmf.

also referred to as the population mean denoted by E 𝑥 or μ equal to the weighted average of the elements x in S where each element is weighted by its respective probability 𝐸 𝑥 = 𝑥 𝑝(𝑥) 𝑥𝜖𝑆

Let x be a discrete rv and 𝑆 = *0, 1, 2, 3+. With the pmf of X given below,

P 𝑥 =

1 4 , 𝑖𝑓 𝑥 𝜖 𝑆 0, 𝑖𝑓 𝑥 ∉ 𝑆 Compute the expected value of X

Let x be a discrete rv and 𝑆 = *1, 2, 3+. With the pmf of X given below,

P 𝑥 =

1 6 𝑥, 𝑖𝑓 𝑥 𝜖 𝑆 0, 𝑖𝑓 𝑥 ∉ 𝑆 Compute the expected value of X

𝐸 𝑥 = 𝑥 𝑝 𝑥 𝑥𝜖𝑆 = 1 16 1 + 2 16 2 + 3 16 3 =^16 +^46 +^96 =^146 𝑜𝑟 2^13

Probability Distributions and Expected Values of Discrete Random Variables, Study notes of Mathematics

Related documents

Partial preview of the text

Download Probability Distributions and Expected Values of Discrete Random Variables and more Study notes Mathematics in PDF only on Docsity!

illustrate a random variable (discrete and

continuous)

distinguish between a discrete and a

continuous random variable

find possible values of a random variable

illustrate probability distributions for discrete

random variable and its properties

An experiment consists of rolling a pair of dice.

Let Y be the random variable representing the

absolute value of the difference of the numbers

that come up. Construct the probability

distribution of Y in tabular, graphical and

formula form.

the values of random variables for which the

probability is greater than zero: 𝑃 𝑥 > 0

*the mass points are: 0, 1, 2, 3, 4 and 5

The possible values (Support S) of a random

variable is the set of values that a random

variable can take. This means that the elements

of S can be put into one-to-one correspondence

with the set of natural numbers.

10 𝐹 𝑥 = 3^ =

10 𝐹 𝑥 = 4^ =