IF arithmetic mean is 10 and V(X)=4
find mean and standard deviation of z where Z=5x+4
To find the mean and standard deviation of Z, we need to know the mean and variance of X. The mean can be directly found using the given information, but the variance of X is required to calculate the standard deviation of Z.
1. Finding the mean of X (μx):
The given information states that the arithmetic mean of X (μx) is 10.
Therefore, the mean of X is μx = 10.
2. Finding the variance of X (V(X)):
The variance of X (V(X)) is given as 4.
Therefore, V(X) = 4.
3. Finding the mean of Z (μz):
The given equation for Z is Z = 5X + 4.
To find the mean of Z (μz), substitute the mean of X into the equation:
μz = 5μx + 4
Plugging in the value of μx = 10:
μz = 5 * 10 + 4
μz = 50 + 4
μz = 54
Therefore, the mean of Z (μz) is 54.
4. Finding the variance of Z (V(Z)):
To find the variance of Z (V(Z)), we need to use the formula for the variance of a linear transformation of a random variable.
V(Z) = (a^2) * V(X)
Substituting the given equation and V(X) value into the formula:
V(Z) = (5^2) * 4
V(Z) = 25 * 4
V(Z) = 100
Therefore, the variance of Z (V(Z)) is 100.
5. Finding the standard deviation of Z (σz):
To find the standard deviation of Z (σz), take the square root of the variance of Z (V(Z)).
σz = √V(Z)
σz = √100
σz = 10
Therefore, the standard deviation of Z (σz) is 10.
In conclusion, the mean of Z (μz) is 54, and the standard deviation of Z (σz) is 10.