# Super Tough Probability Question

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• Nov 25th 2009, 08:25 PM
Porter1
Super Tough Probability Question
........................
• Nov 26th 2009, 12:42 AM
matheagle
Just do (a) directly with polar co-ordinates.
That's how you prove the normal density integrates to one.
• Nov 26th 2009, 08:00 AM
awkward
Quote:

Originally Posted by Porter1
Hi all, I have this question and I really need your help!

(a) Show that Г(1/2) = √(П) (<-Square root of Pi) by writing
Г(1/2) = ∫(0)->(∞) y^(-1/2) e^(-y) dy
by making the transformation y = (x^2)/2 and using the standard normal density.
[snip]

You can work this out from basics as suggested by MathEagle, but if you are allowed to assume some basic knowledge of the Normal distribution there is a shortcut, which I think is probably what you are intended to do--

After you make the substitution, you should have
$\Gamma(1/2) = \sqrt{2} \; \int_0^\infty \exp(-x^2/2) \, dx$
which is equal to
$\sqrt{2} \cdot \sqrt{2 \pi} \cdot \frac{1}{\sqrt{2 \pi}} \; \int_0^{\infty} \exp(-x^2/2) \, dx$
Based on your knowledge of the Normal distribution, you should know the value of $\frac{1}{\sqrt{2 \pi}} \; \int_0^{\infty} \exp(-x^2/2) \, dx$.

Take it from there.
The text contains a mathematical question related to proving a property of the Gamma function using a specific transformation and the standard normal density. An answer or a method to solve the question is provided in the text.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
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- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
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# Super Tough Probability Question

Printable View

• Nov 25th 2009, 08:25 PM
Porter1
Super Tough Probability Question
........................
• Nov 26th 2009, 12:42 AM
matheagle
Just do (a) directly with polar co-ordinates.
That's how you prove the normal density integrates to one.
• Nov 26th 2009, 08:00 AM
awkward
Quote:

Originally Posted by Porter1
Hi all, I have this question and I really need your help!

(a) Show that Г(1/2) = √(П) (<-Square root of Pi) by writing
Г(1/2) = ∫(0)->(∞) y^(-1/2) e^(-y) dy
by making the transformation y = (x^2)/2 and using the standard normal density.
[snip]

You can work this out from basics as suggested by MathEagle, but if you are allowed to assume some basic knowledge of the Normal distribution there is a shortcut, which I think is probably what you are intended to do--

After you make the substitution, you should have
$\Gamma(1/2) = \sqrt{2} \; \int_0^\infty \exp(-x^2/2) \, dx$
which is equal to
$\sqrt{2} \cdot \sqrt{2 \pi} \cdot \frac{1}{\sqrt{2 \pi}} \; \int_0^{\infty} \exp(-x^2/2) \, dx$
Based on your knowledge of the Normal distribution, you should know the value of $\frac{1}{\sqrt{2 \pi}} \; \int_0^{\infty} \exp(-x^2/2) \, dx$.

Take it from there.
user
The text contains a mathematical question related to proving a property of the Gamma function using a specific transformation and the standard normal density. An answer or a method to solve the question is provided in the text.

TYPE: SINGLE_QUESTION_WITH_ANSWER