Financial Risk Manager Part 1

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Explanation:

Conditional Distributions in Probability

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

Recall that given any two events $A$ and $B$ :

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

This result can be applied in bivariate distributions. 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:

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)}

Marginal Distribution of Company B ( $X_2$ )

Company B ( $X_2$ ) Profits	-50 Million	0 Million	10 Million	100 Million
$P(X_2 = x_2)$	0.0712	0.4228	0.3482	0.1583

Joint Distribution of Company A and Company B

Company B ( $X_2$ ) Profits	Company A ( $X_1$ ) Profits
	-1 Million
-50 Million	0.0197
0 Million	0.0390
10 Million	0.011
100 Million	0

Conditional Distribution of Company A Given Company B = 100

To calculate the conditional distribution, divide each entry in the last row by the marginal probability $0.1583$ :

Company A ( $X_1$ ) Profits	-1 Million	0 Million	2 Million	4 Million
$P(X_1	X_2 = 100)$	$0 / 0.1583 = 0$	$0.0309 / 0.1583 = 0.1952$	$0.0656 / 0.1583 = 0.4144$

Explanation:

Conditional Distributions in Probability

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

Recall that given any two events $A$ and $B$ :

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

This result can be applied in bivariate distributions. 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:

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)}

Marginal Distribution of Company B ( $X_2$ )

Company B ( $X_2$ ) Profits	-50 Million	0 Million	10 Million	100 Million
$P(X_2 = x_2)$	0.0712	0.4228	0.3482	0.1583

Joint Distribution of Company A and Company B

Company B ( $X_2$ ) Profits	Company A ( $X_1$ ) Profits
	-1 Million
-50 Million	0.0197
0 Million	0.0390
10 Million	0.011
100 Million	0

Conditional Distribution of Company A Given Company B = 100

To calculate the conditional distribution, divide each entry in the last row by the marginal probability $0.1583$ :

Company A ( $X_1$ ) Profits	-1 Million	0 Million	2 Million	4 Million
$P(X_1	X_2 = 100)$	$0 / 0.1583 = 0$	$0.0309 / 0.1583 = 0.1952$	$0.0656 / 0.1583 = 0.4144$

Comments (1)

Yash ParwaniApr 13, 2026 5:51 AM

There are no tables provided in the question.

LeetQuiz .Apr 13, 2026 7:03 PM

Hi Yash, Thanks for pointing out. We've reviewed and updated the question accordingly. Now, the question has been answerable!

Q.3745 The yearly profits of the two firms A and B can be summarized in the following probability matrix.

Company A (X₁) Profits
-1 Million 0 Million 2 Million 4 Million
Company B (X₂)
Profits
-50 Million 0.0197 0.0395 0.010 0.002
0 Million 0.0390 0.230 0.124 0.0298
10 Million 0.011 0.127 0.144 0.0662
100 Million 0 0.0309 0.0656 0.0618

What is the conditional distribution of company A given that company B made a profit of 100 Million?

A.

Company A(X₁) Profits -1 Million 0 Million 2 Million 4 Million
P(X₁|X₂ = 100) 0.0697 0.4274 0.3436 0.1598

B.

Company A(X₁) Profits -1 Million 0 Million 2 Million 4 Million
P(X₁|X₂ = 100) 0.0697 0.5274 0.6436 0.1598

C.

Company A(X₁) Profits -1 Million 0 Million 2 Million 4 Million
P(X₁|X₂ = 100) 0.0697 0.5274 0.3436 0.2598

D.

Company A(X₁) Profits -1 Million 0 Million 2 Million 4 Million
P(X₁|X₂ = 100) 0 0.1952 0.4144 0.3904

	Company A (X₁) Profits
	-1 Million	0 Million	2 Million	4 Million
Company B (X₂)
Profits
-50 Million	0.0197	0.0395	0.010	0.002
0 Million	0.0390	0.230	0.124	0.0298
10 Million	0.011	0.127	0.144	0.0662
100 Million	0	0.0309	0.0656	0.0618

Company A(X₁) Profits	-1 Million	0 Million	2 Million	4 Million
P(X₁\|X₂ = 100)	0.0697	0.4274	0.3436	0.1598

Company A(X₁) Profits	-1 Million	0 Million	2 Million	4 Million
P(X₁\|X₂ = 100)	0.0697	0.5274	0.6436	0.1598

Company A(X₁) Profits	-1 Million	0 Million	2 Million	4 Million
P(X₁\|X₂ = 100)	0.0697	0.5274	0.3436	0.2598

Company A(X₁) Profits	-1 Million	0 Million	2 Million	4 Million
P(X₁\|X₂ = 100)	0	0.1952	0.4144	0.3904

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Last updated: April 13, 2026 at 19:01

Table A

28.6%

Table B

28.6%

Table C

14.3%

Table D

28.6%

Financial Risk Manager Part 1

Get started today

Conditional Distributions in Probability

Marginal Distribution of Company B (X2X_2X2​)

Joint Distribution of Company A and Company B

Conditional Distribution of Company A Given Company B = 100

Conditional Distributions in Probability

Marginal Distribution of Company B (X2X_2X2​)

Joint Distribution of Company A and Company B

Conditional Distribution of Company A Given Company B = 100

Comments (1)

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Comments (1)

Marginal Distribution of Company B ( $X_2$ )

Marginal Distribution of Company B ( $X_2$ )