# Help with discrete random variables

• sneaky666
In summary, the conversation discusses the probability functions of Z and W, where Z represents the outcome of flipping a coin and W represents a mathematical expression involving Z. The probability function for Z is 1/2 for Z being 1 or 3, and 0 otherwise. The probability function for W is also 1/2 for W being 2 or 12, and 0 otherwise. The second part of the conversation involves computing the probability of Z being between 5 and 9 when Z follows a Geometric distribution with parameter theta.
sneaky666

## Homework Statement

1.
Suppose u flip a coin
Z = 1 if the coin is heads
Z = 3 if the coin is tails
W = Z^2 + Z
a)
what is the probability function of Z?
b)
what is the probability function of W?

2.
Let Z ~ Geometric (theta). Compute P(5<=Z<=9).

## The Attempt at a Solution

1.
what i think
a)
3 if tail
0 otherwise
b)
12 if tail
0 otherwise

I don't see how the probability part comes into this, where do i include the 1/2 chance part...or maybe its:
a)
Pz(Z) = 1/2 if Z is 1 or 3
0 otherwise
b)
Pw(W) = 1/2 if W is 2 or 12
0 otherwise

2.

One of the important things you need to do in math is be precise with your notation. Tell us what is PZ(z) supposed to represent? That's where some of your confusion lies.

## What is a discrete random variable?

A discrete random variable is a type of random variable that can only take on a finite or countably infinite number of values. These values are usually represented by whole numbers, such as the number of heads in a series of coin tosses or the number of red marbles in a bag.

## What is the difference between a discrete random variable and a continuous random variable?

The main difference between a discrete and continuous random variable is that a discrete random variable can only take on specific, separate values while a continuous random variable can take on any value within a given range. For example, the number of children in a family is a discrete random variable while the height of a person is a continuous random variable.

## How do you calculate the mean of a discrete random variable?

To calculate the mean of a discrete random variable, you need to multiply each possible value by its corresponding probability and then add all these products together. The formula for the mean of a discrete random variable is:
Mean = (value1 x probability1) + (value2 x probability2) + ... + (valueN x probabilityN)

## What is the probability distribution of a discrete random variable?

The probability distribution of a discrete random variable is a table, graph, or formula that shows the possible values of the variable and their corresponding probabilities. It is used to determine the likelihood of each outcome occurring in a random experiment.

## What is the difference between a probability mass function and a cumulative distribution function?

A probability mass function (PMF) is a formula or table that gives the probability of a discrete random variable taking on a specific value. On the other hand, a cumulative distribution function (CDF) gives the probability of a discrete random variable being less than or equal to a certain value. In other words, the CDF is the cumulative sum of the PMF.

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