Let X and Y be normal random variables with means 0 and 2, respectively, and variances 1 and 9, respectively. Find the following, using the standard normal table. Express your answers to an accuracy of 4 decimal places.
1. P(X>0.75)=
2. P(X≤−1.25)=
3. Let Z=(Y−3)/4. Find the mean and the variance of Z.
E[Z]=
var(Z)=
4. P(−1≤Y≤2)=
1 P(X>0.75)= 0.2266
2 P(X≤−1.25)= 0.1056
Let Z=(Y−3)/4. Find the mean and the variance of Z.
3. E[Z]= -0.25
4. var(Z)= 0.5625
5. P(−1≤Y≤2)= 0.3413
To find the probabilities using the standard normal table, we first need to standardize the variables using the Z-score formula:
Z = (X - μ) / σ
where Z is the standard normal random variable, X is the original random variable, μ is the mean of the random variable, and σ is the standard deviation of the random variable.
Let's calculate the values step by step:
1. P(X > 0.75):
To find this probability, we need to calculate the area under the standard normal curve to the right of 0.75.
First, standardize X:
Z = (0.75 - 0) / 1 = 0.75
Next, we can use the standard normal table to find the cumulative probability up to Z = 0.75. Look up 0.75 in the table and find the corresponding probability, which is 0.7734.
However, we need to find the probability to the right of 0.75, so we subtract the cumulative probability from 1:
P(X > 0.75) = 1 - 0.7734 = 0.2266
2. P(X ≤ -1.25):
To find this probability, we need to calculate the area under the standard normal curve to the left of -1.25.
First, standardize X:
Z = (-1.25 - 0) / 1 = -1.25
Next, using the standard normal table, find the cumulative probability up to Z = -1.25. Look up -1.25 in the table and find the corresponding probability, which is 0.1056.
P(X ≤ -1.25) = 0.1056
3. Let Z = (Y - 3) / 4.
We are given that Y is a normal random variable with a mean of 2 and a variance of 9.
To find the mean and variance of Z, we need to apply the formula for transforming a random variable using linear transformations.
The mean of Z is given by:
E[Z] = (E[Y] - 3) / 4
Substituting the mean of Y, we get:
E[Z] = (2 - 3) / 4 = -1 / 4 = -0.25
The variance of Z is given by:
var(Z) = (var(Y)) / (4^2)
Substituting the variance of Y, we get:
var(Z) = 9 / 16 = 0.5625
4. P(-1 ≤ Y ≤ 2):
To find this probability, we need to calculate the area under the normal curve between -1 and 2.
First, we standardize the values:
Z1 = (-1 - 2) / 3 = -1
Z2 = (2 - 2) / 3 = 0
Using the standard normal table, find the cumulative probabilities for Z1 and Z2. From the table, we find the probabilities:
P(Z ≤ -1) = 0.1587
P(Z ≤ 0) = 0.5
Finally, to find the probability between -1 and 2, subtract the cumulative probability for Z1 from the cumulative probability for Z2:
P(-1 ≤ Y ≤ 2) = P(Z ≤ 0) - P(Z ≤ -1) = 0.5 - 0.1587 = 0.3413
So, P(-1 ≤ Y ≤ 2) = 0.3413.