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Joint Distributions of R. V.

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- Joint probability distribution function: f(x,y) = P(X=x, Y=y)
- Example Ch 6, 1c, 1d

- Two variables are independent if, for any two sets of real numbers A and B,
- Operationally: two variables are indepndent iff their joint pdf can be “separated” for any x and y:

- The expectation of a sum equals the sum of the expectations:
- The variance of a sum is more complicated:
- If independent, then the variance of a sum equals the sum of the variances

- For any two events, E and F,
- Conditional pdf:
- Examples Ch 6, 4a, 4b

- Conditional cdf:

- Example: what is the probability that the TSX is up, conditional on the S&P500 being up?

- Conditional pdf:
- Conditional cdf:
- Example 5b

- Example: what is the probability that the TSX is up, conditional on the S&P500 being up 3%?

- = joint pdf of X1 and X2
- Equations and can be uniquely solved for and given by:
and

- The functions and have continuous partial derivatives:

- Under the conditions on previous slide,
- Example: You manage two portfolios of TSX and S&P500:
- Portfolio 1: 50% in each
- Portfolio 2: 10% TSX, 90% S&P 500

- What is the probability that both of those portfolios experience a loss tomorrow?

- Example 7a – uniform and normal cases

- Given limited data we make educated guesses about the true parameters
- Estimation of the mean
- Estimation of the variance
- Random sample

- Population parameter describes the true characteristics of the whole population
- Sample parameter describes characteristics of the sample
- Statistics is all about using sample parameters to make inferences about the population parameters

- The sample mean follows a t-distribution:

- We can estimate the mean, but we’d like to know how accurate our estimate is
- We’d like to put upper and lower bounds on our estimate
- We might need to know whether the true mean is above certain value, e.g. zero

- We already know the distribution of our estimate of the mean
- To construct a 95% confidence interval, for instance, just find the values that contain 95% of the distribution

falls in this region 95% of the time

2.5% of the distribution

2.5% of the distribution

Critical values

Critical values

- The critical values are available from a table or in Matlab
>> tinv(.975, n-1)

- If the confidence interval includes zero, then the sample mean is not statistically different from the population mean we are testing
- One-sided vs. two-sided tests

- Are the returns on the S&P 500 significantly above zero?
- Sample mean = .23
- Sample standard deviation = .59
- Sample size = 128

- Compute the test:
- At 95% the critical value is 1.98
- Therefore, we reject that the returns are zero

- The direct use of historical data requires the following assumptions:
- The true distribution of returns is constant through time and will not change in the future
- Each period represents an independent draw from this distribution

GNP Growth

Spread

Regression Line:

GNP Growth

Spread

- Minimize the squared residuals:

- Regression equation:
- Minimize the squared residuals: