Conditional normal probability, repost

In summary, the problem involves finding the probability that family A will receive at least twice as much sunshine as family B during their respective stays at a resort. To solve this, the distribution of the random variable A - 2B is considered, as it represents the difference in sunshine hours between the two families. By using properties of normally distributed variables, the probability can be calculated.
  • #1
Gauss M.D.
153
1

Homework Statement



The number of hours of sunshine in one week in a specific resort is assumed to follow a normal distribution with expectation 43 and standard deviation 17.

Family A will spend the first three weeks of the summer at the resort. Family B will spend the LAST two weeks of the summer at the resort.

Assuming everything is independent, what is the probability that family A will get atleast twice as much sunshine as family B?

Homework Equations



A = # of sunshine hours for family A
B = # of sunshine hours for family B

The Attempt at a Solution



E(A) = 3*E(X) = 129
σ(A) = sqrt(3)*σ(X) = 29.4

E(B) = 86
σ(B) = sqrt(2)*σ(X) = 24

So A ~ N(129, 29.4), B ~ N(86, 24)

If we let Z = A/B and try to find P(Z > 2), we do not have a linear combination of normally distributed random variables so I guess that's not the right way to go, assuming we want to do the normal table reading thing. So I don't know where to go from here. Help!
 
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  • #2
You could try an integration over A or B, and see if that works.

Edit: Oh, I did not see this. Follow awkward's hint, that is way easier.
 
Last edited:
  • #3
Hint: What can you say about the distribution of the random variable A - 2B?
 
  • #4
awkward said:
Hint: What can you say about the distribution of the random variable A - 2B?

D'oh! Thank you so much!
 

1. What is conditional normal probability?

Conditional normal probability is a statistical concept that calculates the probability of an event occurring based on a set of given conditions or criteria. It is typically used in situations where the outcome of an event is affected by one or more variables.

2. How is conditional normal probability calculated?

Conditional normal probability is calculated using the formula P(A|B) = P(A∩B)/P(B), where P(A|B) represents the probability of event A occurring given that event B has already occurred, P(A∩B) represents the joint probability of event A and B occurring, and P(B) represents the probability of event B occurring.

3. What is the difference between conditional normal probability and regular normal probability?

The main difference between conditional normal probability and regular normal probability is that conditional normal probability takes into account a specific set of conditions or criteria, while regular normal probability does not. Conditional normal probability is used to calculate the probability of an event occurring given that certain conditions are met, while regular normal probability is used to calculate the probability of an event occurring without any specific conditions.

4. What are some real-life applications of conditional normal probability?

Conditional normal probability is commonly used in fields such as finance, economics, and social sciences to predict future outcomes based on specific conditions. It is also used in medical research to evaluate the effectiveness of treatments based on different variables and in engineering to assess the reliability of systems based on various factors.

5. What are the limitations of conditional normal probability?

One limitation of conditional normal probability is that it assumes that the variables are normally distributed. In situations where the variables are not normally distributed, the results may not accurately reflect reality. Additionally, the accuracy of the results may be affected by the quality and quantity of data used in the calculations.

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