Dependent random variables

1. Sablon:Matematika függő valószínűségi változók

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Dependent random variables are random variables whose outcomes or values influence each other in some way. This means that the probability distribution of one variable is affected by the values of the others. In contrast to independent random variables, where knowing the outcome of one gives no information about the other, dependent variables exhibit some kind of relationship, which could be direct or indirect.

Key Concepts

1. conditional probability: If random variables $X$ and $Y$ are dependent, the probability of $Y$ given $X$ is not equal to the marginal probability of $Y$. Instead, it is calculated using conditional probability: $$P(Y|X)\ne P(Y)$$ The distribution of $Y$ depends on the value of $X$.

2. Covariance and Correlation: Dependence between two random variables can be quantified using covariance or correlation. If $X$ and $Y$ have a non-zero covariance, they are likely dependent: $$\text{Cov}(X,Y)=E[(X-E[X])(Y-E[Y])]$$ Similarly, the correlation coefficient can indicate the strength and direction of the dependence.

3. Joint Probability Distribution: For dependent variables, the joint probability distribution $P(X,Y)$ cannot be factored into the product of the individual marginal distributions $P(X)$ and $P(Y)$: $$P(X,Y)\ne P(X)P(Y)$$ This contrasts with independent variables where this equality holds.

4. Bayesian Networks: A graphical model that shows dependencies between random variables using directed acyclic graphs (DAGs). Dependencies can be captured through conditional relationships between variables.

Examples

- Weather and Umbrella Sales: The random variables "rain" and "umbrella sales" are dependent. If it rains, umbrella sales increase, and the outcome of one variable influences the other.

- Stock Prices: The prices of stocks in the same sector tend to be dependent. The performance of one company may affect others in the same industry.

Understanding the dependencies between random variables is crucial in fields like statistics, machine learning, and economics, where predictive modeling often relies on the relationships between variables. Sablon:Engl