Financial Risk Manager Part 1

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The conditional distributions describe the probability of an outcome of a random variable conditioned on the other random variable taking a particular value.

Recall that given any two events A and B, then:

P(A | B) = \frac{P(A \cap B)}{P(B)}

This result can be applied in bivariate distributions. That is, the conditional distribution of $X_1$ given $X_2$ is defined as:

f_{(X_1 | X_2)}(x_1 | X_2 = x_2) = \frac{f_{(X_1, X_2)}(x_1, x_2)}{f_{X_2}(x_2)}

So, in this case, we need:

f_{(X_1 | X_2)}(x_1 | X_2 = 100) = \frac{f_{(X_1, X_2)}(x_1, x_2)}{f_{X_2}(X_2 = 100)}

The marginal distribution of company B ( $f_{X_2}(x_2)$ ) given by:

Company B(X₂) Profits	-50 Million	0 Million	10 Million	100 Million
P(X₂ = x₂)	0.0712	0.4228	0.3482	0.1583

Which table shows the correct conditional distribution of Company A's profits given that Company B's profits are 100 million?

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Table A

Table B

Table C

Table D

Explanation:

The correct answer is D (Table D).

Explanation:

The conditional distribution of Company A's profits given that Company B's profits are 100 million is calculated using the formula:

$f_{(X_1 | X_2)}(x_1 | X_2 = 100) = \frac{f_{(X_1, X_2)}(x_1, x_2)}{f_{X_2}(X_2 = 100)}$

Where:

$f_{(X_1, X_2)}(x_1, x_2)$ is the joint probability
$f_{X_2}(X_2 = 100) = 0.1583$ is the marginal probability of Company B having 100 million profits

The table shows:

For X₁ = -1 Million: P(X₁ | X₂ = 100) = 0
For X₁ = 0 Million: P(X₁ | X₂ = 100) = 0.1952
For X₁ = 2 Million: P(X₁ | X₂ = 100) = 0.4144
For X₁ = 4 Million: P(X₁ | X₂ = 100) = 0.3904

These conditional probabilities sum to 1 (0 + 0.1952 + 0.4144 + 0.3904 = 1.0000), which confirms this is a valid conditional probability distribution._

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