Respuesta :
Answer:
Refer below
Step-by-step explanation:
a) a and b are the lower and higher values of the interval for which uniform distribution is defined.
Here a= 6 and b =10
b) Mean of the uniform distribution= (a+b)/2 = (6+10)/2 =8
Or int x (1/4) dx = x^2/8 = 8
c) Variance of the uniform distribution = (b^2-a^2)/12 = (100-64)/12
= 36/12 =3
Std dev = sq rt of 3 = 1.732
d) To find total area
PDF of the distribution = 1/(b-a) = 1/4, 6<x<10
Area = \int 6 to 10 of 1/4 dx
= x/4
Subtitute limits
= (10-6)/4 =1
So total area = 1
d)P(X>7) = int 7 to 10 of 1/4 dx = 3/4
e) P(7<x<9) = Int 7 to 9 of 1/4 dx = 2/4 = 1/2
Probabilities are used to determine the chances of events
The values of a and b
The interval is given as 6 to 10.
So, the values of a and b are 6 and 10, respectively.
The mean
For a uniform distribution, the mean is:
[tex]\bar x = \frac{a + b}{2}[/tex]
So, we have:
[tex]\bar x = \frac{6 + 10}{2}[/tex]
[tex]\bar x = 8[/tex]
Hence, the mean is 8
The standard deviation
For a uniform distribution, the standard deviation is:
[tex]\sigma = \sqrt{\frac{b^2 - a^2}{12}}[/tex]
So, we have:
[tex]\sigma = \sqrt{\frac{10^2 - 6^2}{12}}[/tex]
[tex]\sigma = 2.31[/tex]
Hence, the standard deviation is 2.31
The total area
This is calculated as:
[tex]A = \int\limits^b_a \frac{1}{b - a}\ dx[/tex]
So, we have:
[tex]A = \int\limits^{10}_6 \frac{1}{10 - 6}\ dx[/tex]
[tex]A = \int\limits^{10}_6 \frac{1}{4}\ dx[/tex]
Factor out 1/4
[tex]A = \frac{1}{4}\int\limits^{10}_6 \ dx[/tex]
Integrate
[tex]A = \frac{1}{4} * x|\limits^{10}_6[/tex]
Expand
[tex]A = \frac{1}{4} * (10 - 6)[/tex]
[tex]A = 1[/tex]
Hence, the total area of the distribution is 1.00
The probability of a value more than 7
This is calculated as:
[tex]P(x > 7) = \frac{1}{4} * x|\limits^{10}_7[/tex]
Expand
[tex]P(x > 7) = \frac{1}{4} * (10 - 7)[/tex]
[tex]P(x > 7) = 0.75[/tex]
Hence, the probability of a value more than 7 is 0.75
The probability of a value between 7 and 9
This is calculated as:
[tex]P(7 < x < 9) = \frac{1}{4} * x|\limits^9_7[/tex]
Expand
[tex]P(7 < x < 9) = \frac{1}{4} * (9 - 7)[/tex]
[tex]P(7 < x < 9) = 0.50[/tex]
Hence, the probability of a value between 7 and 9 is 0.50
Read more about probability at:
https://brainly.com/question/25870256
