Getting the probability distribution of a random variable

In summary, obtaining the probability distribution of a random variable involves identifying all possible outcomes and their associated probabilities. This can be achieved through methods such as empirical observation, theoretical models, or statistical inference. The distribution can be discrete, where outcomes are countable, or continuous, where outcomes form a range. Key concepts include probability mass functions for discrete variables and probability density functions for continuous variables, which together provide a complete description of the random variable's behavior and facilitate further statistical analysis.
  • #1
rcktbr
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Homework Statement
Described below
Relevant Equations
Marginal Probability Equation
X and Y are discrete random variables with the following joint distribution:
1723338968054.png

a) Calculate the probability distribution, mean, and variance of Y.

My attempt:

I have calculated the probability for different values of Y and X using the following equation: ##\text{Pr(Y = y)}## = ##\sum_{i=1}^l## ##\rm{Pr(X = x_i, Y = y)}##. I have arrived at some results, such as ##\text{Pr(Y = 14)} = 0.21##, ##\text{Pr(Y = 22)} = 0.23##, ##\text{Pr(Y = 30)} = 0.30##, ##\text{Pr(Y = 40)} =0.15## and ##\text{Pr(Y = 65)} = 0.11##.

However, I wanted to know how the formula above could be used in calculating the probabilities. What would be ##x_1##? What to put in i in an equation such as ##\text{Pr(Y = 14)}## = ##\sum_{i=?}^?## ##\rm{Pr(X = x_i, Y = 14)}##?
 
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  • #2
With ##i## as a subscript of ##x## values, it should be ##i=1,2,3##. You can then assign the values of the ##x_i##s in whatever order you want. I would start from the top row to the bottom. ##x_1=1,\ x_2=5,\ x_3=8##.
 
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  • #3
Alright. I will do the same for Y so. Thanks.
 

FAQ: Getting the probability distribution of a random variable

What is a probability distribution?

A probability distribution is a mathematical function that describes the likelihood of different outcomes for a random variable. It provides a complete description of the probabilities associated with all possible values that the random variable can take.

How do I determine the probability distribution of a random variable?

To determine the probability distribution of a random variable, you can use empirical data to estimate probabilities or theoretical models based on assumptions about the underlying process. For discrete variables, you can create a probability mass function (PMF), while for continuous variables, you can use a probability density function (PDF).

What is the difference between discrete and continuous probability distributions?

Discrete probability distributions are used for random variables that can take on a countable number of distinct values, such as the roll of a die. Continuous probability distributions, on the other hand, are used for random variables that can take on an infinite number of values within a given range, such as the height of individuals.

What are some common types of probability distributions?

Some common types of probability distributions include the binomial distribution, Poisson distribution, normal distribution, uniform distribution, and exponential distribution. Each has unique properties and is applicable in different scenarios depending on the nature of the random variable.

How can I visualize a probability distribution?

You can visualize a probability distribution using graphs such as histograms for discrete distributions or probability density plots for continuous distributions. These visualizations help in understanding the shape, spread, and central tendency of the distribution.

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