Problem with finding mathematical expectation in normal distribution

[JujiGatame]

Hello everyone!

I am trying to solve this normal distribution task but I can't find mathematical expectation.

Here is the task:

The highest temperature in the Fahrenheit in one day in July in Antarctica corresponds to a Gaussian random variable with a variance of 225 F. With a probability of 0.5, the temperature T is greater than 10 degrees. What is the probability that the temperature is higher than 32 degrees?

Here is picture where I stuck

 

[Cubist]

We don't know μ. But, can you tell me what is the probability P(T > μ) ?

EDIT: My favourite is sankaku jime ?

 

[JujiGatame]

Maybe this will be clearer

Here i just need find sigma or, like in first task mathematical expectation. Somehow i need to use this table to get the correct value but I don't know how

 

[JujiGatame]

We don't know μ. But, can you tell me what is the probability P(T > μ) ?

EDIT: My favourite is sankaku jime ?

I'm not sure if I understand. I just need to find the solution. Like in this task, I need to find sigma from this last formula. I have table value of standrad normal distribution and I need to use it to get the value of sigma

 

[Cubist]

...can you tell me what is the probability P(T > μ) ?

This is a hint. In other words, can you tell me the probability that the temperature will be greater than the mean value, which also corresponds to z=0 ?

 

[Cubist]

... I need to find sigma from this last formula.

You question very nearly states the value of sigma (see the bit highlighted red)...

The highest temperature in the Fahrenheit in one day in July in Antarctica corresponds to a Gaussian random variable with a variance of 225 F. With a probability of 0.5, the temperature T is greater than 10 degrees. What is the probability that the temperature is higher than 32 degrees?

Search for "variance" on this Wikipedia page about the normal distribution to find the relationship between variance and sigma ( σ ).

 

[JujiGatame]

You question very nearly states the value of sigma (see the bit highlighted red)...


Search for "variance" on this Wikipedia page about the normal distribution to find the relationship between variance and sigma ( σ ).

I know that V[X]=( σ )^2 is variance and E[X]= μ mathematcial expectation. I know that I have complicated all this thing and I'm sorry. Neglect the first post that I have sent. I just need to know from two last pictures how to get σ . So from this first picture I need to solve last equation (I need to get σ ) and I must use this normal distribution table. How can I use table if I don't get proper Ø (there is σ that I need to know to get Ø).

 

[Cubist]

...I know that I have complicated all this thing and I'm sorry. Neglect the first post that I have sent.

OK, sorry, I'll ignore the first post...

I just need to know from two last pictures how to get σ . So from this first picture I need to solve last equation (I need to get σ ) and I must use this normal distribution table. How can I use table if I don't get proper Ø (there is σ that I need to know to get Ø).

So you have...

[MATH] \Phi\left( \frac{18}{\sigma} \right) - \Phi\left( \frac{-18}{\sigma} \right) = 0.997 [/MATH]
Using [MATH] \Phi(-x)=1-\Phi(x) [/MATH] (if you need convincing, then draw a sketch of the PDF and shade the two areas)

[MATH] \Phi\left( \frac{18}{\sigma} \right) - \left(1-\Phi\left( \frac{18}{\sigma} \right) \right)= 0.997 [/MATH]
...continue to obtain \( \Phi\left( \frac{18}{\sigma} \right) \) = <something> and then you can perform a reverse lookup in the table

 

[JujiGatame]

OK, sorry, I'll ignore the first post...


So you have...

[MATH] \Phi\left( \frac{18}{\sigma} \right) - \Phi\left( \frac{-18}{\sigma} \right) = 0.997 [/MATH]
Using [MATH] \Phi(-x)=1-\Phi(x) [/MATH] (if you need convincing, then draw a sketch of the PDF and shade the two areas)

[MATH] \Phi\left( \frac{18}{\sigma} \right) - \left(1-\Phi\left( \frac{18}{\sigma} \right) \right)= 0.997 [/MATH]
...continue to obtain \( \Phi\left( \frac{18}{\sigma} \right) \) = <something> and then you can perform a reverse lookup in the table

The result is σ = 6.06. I have used the supstituton method and it works. But what if I have one number with 18/σ, and other with 15/σ (just another example). Then I have x and y and only one equation x+y=1.997 . So how would I solve that?

 

[Cubist]

The result is σ = 6.06.

Looks good, well done!

But what if I have one number with 18/σ, and other with 15/σ (just another example). Then I have x and y and only one equation x+y=1.997 . So how would I solve that?

There's no easy/ direct way that I'm aware of.

If you have to use the table then I recommend doing a binary search (click) by hand. This would involve repeatedly calculating phi(18/σ) + phi(15/σ) for different σ to home in on the value that yields 1.997

If you can use a computer then you could produce a speadsheet or program which would find a very accurate answer.

 

[JujiGatame]

Looks good, well done!


There's no easy/ direct way that I'm aware of.

If you have to use the table then I recommend doing a binary search (click) by hand. This would involve repeatedly calculating phi(18/σ) + phi(15/σ) for different σ to home in on the value that yields 1.997

If you can use a computer then you could produce a speadsheet or program which would find a very accurate answer.

Thank you man. I really appreciate your help. I would buy you a beer, but I can't. So,, cheers and thank you once again. ? ? ?