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.

P(X>0.75)=
P(X≤−1.25)=
Let Z=(Y−3)/4. Find the mean and the variance of Z.

E[Z]=
var(Z)=
P(−1≤Y≤2)=

First two questions and last.

Z = (score-mean)/SD

SD^2 = variance

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability of the Z score.

P(X>0.75)= 0.2266

P(X≤−1.25)= 0.1056

Can't figure out these:
E[Z]=
var(Z)=
P(−1≤Y≤2)=

anyone?

P(X>0.75)= 0.2266

P(X≤−1.25)= 0.1056
E[Z]= -1/4
var(Z)= 9/16
P(−1≤Y≤2)= ?

P(−1≤Y≤2)= 0.3413

To find the probabilities using the standard normal table, we need to standardize the random variables by subtracting the mean and dividing by the standard deviation.

For X:
Mean = 0
Variance = 1
Standard Deviation = √(Variance) = √1 = 1

To find P(X > 0.75), we need to standardize 0.75:
Z = (X - Mean) / Standard Deviation = (0.75 - 0) / 1 = 0.75

Now, we look up the probability associated with Z = 0.75 in the standard normal table. The table shows the area to the left of the given Z-value. But since we need P(X > 0.75), we subtract the table value from 1:

P(X > 0.75) = 1 - P(X ≤ 0.75)

By looking up Z = 0.75 in the standard normal table, we find that P(X ≤ 0.75) is 0.7734. Therefore:

P(X > 0.75) = 1 - 0.7734 = 0.2266

To find P(X ≤ -1.25), we standardize -1.25:
Z = (X - Mean) / Standard Deviation = (-1.25 - 0) / 1 = -1.25

We find the probability associated with Z = -1.25 in the standard normal table, which is 0.1056:

P(X ≤ -1.25) = 0.1056

Next, let's calculate the mean and variance of Z:

Given:
Y has mean = 2
Y has variance = 9

Let Z = (Y - 3) / 4

Mean of Z:
E[Z] = (mean of Y - 3) / 4 = (2 - 3) / 4 = -1/4 = -0.25

Variance of Z:
var(Z) = (variance of Y) / (standard deviation of Z)^2 = 9 / (4^2) = 9 / 16 = 0.5625

Therefore,
E[Z] = -0.25
var(Z) = 0.5625

Now, let's find P(-1 ≤ Y ≤ 2):
First, we standardize -1 and 2:
Standardize -1: Z = (Y - Mean) / Standard Deviation = (-1 - 2) / 3 = -3/3 = -1
Standardize 2: Z = (Y - Mean) / Standard Deviation = (2 - 2) / 3 = 0

Using the standard normal table, we find the probability associated with Z = -1 and Z = 0.
P(Z ≤ -1) = 0.1587
P(Z ≤ 0) = 0.5

Therefore,
P(-1 ≤ Y ≤ 2) = P(Z ≤ 0) - P(Z ≤ -1) = 0.5 - 0.1587 = 0.3413