Suppose x is random variable with normal distribution (u,sigma^2) distribution and p(x<=0)=1/3 and p(x<=1)=2/3 what are the values of u and sigma? Please show me the steps to solve such a question
Thank you
the z-score is the number of standard deviations (σ's) that a value lies from the mean
the z-score table gives the probability of a score's position in a normal distribution
eg. the table gives the mean (z-score, zero) a probability of .5
... half of the population lies below the mean
p = 1/3 for x<=0
... from the table ... zero is .43 σ below the mean
p(x<=1)=2/3
... from the table ... 1 is .43 σ above the mean
Thanks for the answer but I still dont get it how did you get the 0.43?
http://davidmlane.com/hyperstat/z_table.html
click scond circle
put in area = .3333
and area = .6667
and you will get that -.43 and + .43
which means the mean is halfway between 0 and 1 :)
now if the distance from 0 to 1 is 2*.43 sigma, what is sigma?
Thanks alot I have tried coming up algebrically with 0.43 but still no luck there is formula like x-u/sigma but still cant use it any ideas?
To find the values of u (mean) and σ (standard deviation) for the normal distribution, we need to use the properties of the standard normal distribution.
Step 1: Convert the given probabilities to z-scores
Since we are given probabilities for x, we need to convert them to their corresponding z-scores using the standard normal distribution table or a statistical software.
For p(x ≤ 0) = 1/3:
To find the z-score for x = 0, we can use the formula:
z = (x - u) / σ
Substituting x = 0 and p(x ≤ 0) = 1/3, we have:
1/3 = P(x ≤ 0) = P(z ≤ (0 - u) / σ)
Similarly, for p(x ≤ 1) = 2/3:
2/3 = P(x ≤ 1) = P(z ≤ (1 - u) / σ)
Step 2: Solve for the z-scores
We have two equations with two unknowns (u and σ). Let's solve them simultaneously.
From the equation with p(x ≤ 0) = 1/3, we have:
1/3 = P(z ≤ -u / σ) ---- (equation 1)
From the equation with p(x ≤ 1) = 2/3, we have:
2/3 = P(z ≤ (1 - u) / σ) ---- (equation 2)
Step 3: Use the standard normal distribution table or software
Using a standard normal distribution table, we can find the corresponding z-scores for the given probabilities:
For p(z ≤ -u / σ), we find the z-score that corresponds to a cumulative probability of 1/3.
For p(z ≤ (1 - u) / σ), we find the z-score that corresponds to a cumulative probability of 2/3.
Step 4: Determine the values of u and σ
Once we have the z-scores from Step 3, we can equate them with the given equations and solve for u and σ.
For equation 1:
1/3 = P(z ≤ -u / σ)
Find the z-score which corresponds to a cumulative probability of 1/3 and equate it to -u / σ
For equation 2:
2/3 = P(z ≤ (1 - u) / σ)
Find the z-score which corresponds to a cumulative probability of 2/3 and equate it to (1 - u) / σ
Solving these equations will give us the values of u and σ for the given normal distribution.