Let X have the pmf p(x)=((1)/(2))^x, x=1,2,3, …, zero elsewhere. Find the pmf of Y=X^3.

Name: Problem 8, Chapter 1: Introduction to Mathematical Statistics
Uploaded: 2021-07-28T07:17:51-08:00
Duration: 1 min 45 s
Channel: Ahmad Reda
Description: Let X have the pmf p(x)=((1)/(2))^x, x=1,2,3, …, zero elsewhere. Find the pmf of Y=X^3.

step 1 In problem 8 we have the probability mass function for the random variable x which equals half t

Let X have the pmf p(x)=((1)/(2))^x, x=1,2,3, …, zero...

Key Concepts

One-to-One Transformations

A one-to-one transformation is a function that assigns each value of the original random variable to a unique value of the transformed variable. In such cases, the probability of an outcome in the transformed variable can be directly determined from the probability of the corresponding outcome in the original variable using the inverse of the transformation.

Support of a Random Variable

The support of a random variable is the set of all values where the pmf (or probability density function in the continuous case) is non-zero. It is crucial in transformation problems because the transformation function maps the support of the original random variable to the support of the new random variable.

Transformation of Discrete Random Variables

This concept involves finding the distribution of a new random variable defined as a function of an existing random variable. When the transformation is applied, the new variable's pmf can often be derived by relating its outcomes to the outcomes of the original variable using the transformation function.

Probability Mass Function (pmf)

A probability mass function assigns a probability to each value of a discrete random variable, ensuring that the sum over the support of the variable is equal to 1. It is essential in determining the likelihood of each outcome in problems involving discrete random variables.

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