Discrete bivariate distributions

Univariate Distributions

• Started with a table of x and probability of x.
• Answered questions about probabilities, expected value, and variance.
• Introduced binomial, Poisson, and hypergeometric distributions.
• These distributions have special formulas.
• Binomial Distribution: $x$ has a binomial distribution with parameters $n$ and $y$, where $y$ is the probability of success.
• Poisson Distribution: $x$ is a Poisson distribution with parameter $\lambda$ (average).
• Hypergeometric Distribution: Parameters are $n$, $N$, and $e$.
• These distributions are used to create probability tables.
• Univariate refers to having only one $x$ variable.
• Example table:
  • x values: 0, 1, 2
  • Probabilities: 0.1, 0.5, 0.4 (adds up to 1)
• This table is used for calculating expected values.

Bivariate Distributions

• Dealing with both $x$ and $y$ variables.
• Using a table to represent probabilities.
• Table structure:
  • $x$ values (e.g., 0, 1, 2)
  • $y$ values (e.g., 0, 1, 2)
  • Joint probabilities within the table.
• Joint Probabilities: Probabilities of both $x$ and $y$ occurring together (e.g., $P(X=0, Y=0)$).
• Marginal Probabilities: Totals obtained by adding rows or columns of the joint probability table.
• Marginal Probability of X: Summing the joint probabilities across all values of Y for each value of X.
• Marginal Probability of Y: Summing the joint probabilities across all values of X for each value of Y.
• Double Summation: Summing over all $x$ and $y$ values in the table.

Joint Probability

• The value inside the table represents the joint probability.

Marginal Probability

• Adding values in the rows and columns to get the marginal probability $P(Y)$
• The total from adding the probability in the column give the probability of $X$.

Expected Values

• Interested in the expected value of $x$ (mean of $x$) and the expected value of $y$ (mean of $y$).
• Symbols: $\mu<em>x$ and $\mu</em>y$.
• The formula for expected value is similar, focusing on marginal distributions.

Formula for Expected value of X

$\sum<em>{i=1}^{3} x</em>i * P(x_i)$

• Dealing with the X's, calculate by multiplying the X values by their probabilities, resulting in the expected value.

Formula for Expected value of Y

$\sum<em>{i=1}^{3} y</em>i * P(y_i)$

• The Y values are multiplied to get the final answer $0.5$

Variance

• ( \sigma^2_x ) represents the variance of $x$.
• Formula:
  • $\sum<em>{i=1}^{3} x</em>i^2 * P(x_i) - (E[x])^2$
• Calculate $\sum<em>{i=1}^{3} x</em>i^2 * P(x_i)$

Explanation

So, whenever you see expected value, is just the average of the data so the probability distribution will replace that data set so the average of $X$ and $Y$ would be, for example, dealing with phone calls the average number of phone calls is what we're calculating.

• The variance of $y$ uses a similar formula.
• Formulas may seem overwhelming, but they have repeating elements and different symbols.
• Covariance will be similar but include both $x$ and $y$.
• Marginal distributions are used, not the insight of the distribution (to be used for covariance).
• Variance values:
  • Variance of $x$: 0.41
  • Variance of $y$: 0.45
• Example: $x$ and $y$ represent the number of houses sold by two people.
• Probabilities are relative frequencies used to calculate the average number of houses sold.

Covariance

• Covariance measures the relationship between $x$ and $y$, denoted as $\sigma_{xy}$.
• Formula:
  • $\sum<em>{x} \sum</em>{y} x<em>i y</em>i P(x<em>i, y</em>i) - \mu<em>x \mu</em>y$
• Double summation is used because of rows and columns in the table.
• I and j are used as indexes for rows and columns.
• Instead of $x^2$ or $y^2$, we have the product of $x<em>i * y</em>i$.
• Multiply with the joint probability $P(x<em>i, y</em>i)$.
• Subtract $\mu<em>x * \mu</em>y$
• The formula is similar to the variance formula, with the product of $x$ and $y$.
• Expected values and probabilities are used from the table.
• Long summation is needed, summing all values in the table.
• The table should not be bigger than the stated size.
• Interpretation: Negative correlation indicates an inverse relationship between the variables.

Calculation

• X is zero multiplied by zero, multiply by probability = 0.12
• In the next row, it could be one so we add the one multiplied by zero.
• Question will newer give you a table bigger than this.

Coefficient of Correlation

• Represents the correlation coefficient between $x$ and $y$, denoted as $\rho$.
• Formula:
  • $\rho = \frac{\sigma<em>{xy}}{\sqrt{\sigma</em>x \sigma_y}}$
• Values are between -1 and +1.
• Provides a better interpretation than covariance.
• Negative correlation indicates an inverse relationship.
• The closer to zero, the weaker the relationship. A correlation near zero indicates near randomness. And correlation near -1 or 1 indicates a strong inverse or direct correlation.
• If it is closer to zero, it is a weak correlation.
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