Answer:
a) the probability of (2 < X < 6) is 0.6247
b) the value of c is 3.878
c) the value of E[ x² ] is 13
Step-by-step explanation:
Given that;
mean μ = 3
variance = 4
standard deviation s = √variance = √4 = 2
(a) Find the probability P(2 < X < 6)
P(2 < X < 6) = p( (x - μ / s ) < z < (x - μ / s ) )
= p( (2 - 3 / 2 ) < z < (6 - 3 / 2 ) )
= p( -0.5 < z < 1.5)
from z-score table, 1.5; z = 0.9332 and -0.5; z = 0.3085
so
P(2 < X < 6) = 0.9332 - 0.3085 = 0.6247
Therefore, the probability of (2 < X < 6) is 0.6247
b) Find the value c such that P(X > c) = 0.33
with p-value = 0.33, the corresponding z -score to the right is 0.439
we know that;
z = x - μ / s
we substitute
0.439 = x - 3 / 2
x - 3 = 2 × 0.439
x - 3 = 0.878
x = 0.878 + 3
x = 3.878
Therefore, the value of c is 3.878
c) Find E[ x² ].
Variance = E[ x² ] - [ mean ]²
E[ x² ] = Variance + [ mean ]²
we substitute
E[ x² ] = 4 + [ 3 ]²
E[ x² ] = 4 + 9
E[ x² ] = 13
Therefore, the value of E[ x² ] is 13
The distribution follows a normal distribution.
The given parameters are:
[tex]\mathbf{\mu = 3}[/tex] --- mean
[tex]\mathbf{\sigma^2= 4}[/tex] --- variance
(a) P(2 < x < 6)
Start by calculating the standard deviation
[tex]\mathbf{\sigma = \sqrt{\sigma^2}}[/tex]
This gives
[tex]\mathbf{\sigma = \sqrt{4}}[/tex]
[tex]\mathbf{\sigma =2}[/tex]
Calculate the z-scores for x = 2 and 6
[tex]\mathbf{z = \frac{x - \mu}{\sigma}}[/tex]
So, we have:
[tex]\mathbf{z = \frac{2 - 3}{2} = -0.5}[/tex]
[tex]\mathbf{z = \frac{6 - 3}{2} = 1.5}[/tex]
So, the probability becomes
[tex]\mathbf{P(2 < x < 6) = P(-0.5 < z < 1.5)}[/tex]
Using z table of probabilities, we have:
[tex]\mathbf{P(2 < x < 6) =0.9332 - 0.3085}[/tex]
[tex]\mathbf{P(2 < x < 6) =0.6247}[/tex]
(b) Calculate c if P(X > c) = 0.33
Start by calculating the z-score for p-value = 0.33
From the z table, z = 0.439 when p-value = 0.33
Recall that:
[tex]\mathbf{z = \frac{x - \mu}{\sigma}}[/tex]
So, we have:
[tex]\mathbf{0.439 = \frac{x - 3}{2}}[/tex]
Multiply both sides by 2
[tex]\mathbf{0.878= x - 3}[/tex]
Add 3 to both sides
[tex]\mathbf{3.878= x}[/tex]
Rewrite as:
[tex]\mathbf{x = 3.878}[/tex]
Replace x with c
[tex]\mathbf{c = 3.878}[/tex]
The value of c is 3.878
(c) Calculate E[x²]
The variance of a dataset is:
[tex]\mathbf{\sigma^2 =E(x^2) - \mu^2}[/tex]
Substitute known values
[tex]\mathbf{4 =E(x^2) - 3^2}[/tex]
[tex]\mathbf{4 =E(x^2) - 9}[/tex]
Add 9 to both sides
[tex]\mathbf{13 =E(x^2) }[/tex]
Rewrite as:
[tex]\mathbf{E(x^2) = 13 }[/tex]
Hence, the value of [tex]\mathbf{E(x^2) }[/tex] is 13
Read more about normal random variables at:
https://brainly.com/question/11395972
