1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
beks73 [17]
3 years ago
8

Based on the number of voids, a ferrite slab is classified as either high, medium, or low. Historically, 5% of the slabs are cla

ssified as high, 85% as medium, and 10% as low. A group of 20 slabs that are independent regarding voids is selected for test ing. Let X, Y, and Z denote the number of slabs that are classified as high, medium, and low, respectively.
(a) What are the name and values of the parameters of the joint probability distribution of X, Y , and
Z?
(b) What is the range of the joint probability distribution of X, Y, and Z?
(c) What are the name and the values of the parameters of the marginal probability distribution of X?
(d) Determine E[X] and Var(X).
Determine the following:
(e) P{X = 1, Y = 17, Z = 3}
(f) P{X ? 1, Y = 17, Z = 3}
(g) P{X ? 1}
(h) E[Y ]
Mathematics
1 answer:
AnnyKZ [126]3 years ago
8 0

Answer:

(a) Name: Multinomial distribution

Parameters: p_1 = 5\%   p_2 = 85\%   p_3 = 10\%  n = 20

(b) Range: \{(x,y,z)| x + y + z=20\}

(c) Name: Binomial distribution

Parameters: p_1 = 5\%      n = 20

(d)\ E(x) = 1   Var(x) = 0.95

(e)\ P(X = 1, Y = 17, Z = 3) = 0

(f)\ P(X \le 1, Y = 17, Z = 3) =0.07195

(g)\ P(X \le 1) = 0.7359

(h)\ E(Y) = 17

Step-by-step explanation:

Given

p_1 = 5\%

p_2 = 85\%

p_3 = 10\%

n = 20

X \to High Slabs

Y \to Medium Slabs

Z \to Low Slabs

Solving (a): Names and values of joint pdf of X, Y and Z

Given that:

X \to Number of voids considered as high slabs

Y \to Number of voids considered as medium slabs

Z \to Number of voids considered as low slabs

Since the variables are more than 2 (2 means binomial), then the name is multinomial distribution

The parameters are:

p_1 = 5\%   p_2 = 85\%   p_3 = 10\%  n = 20

And the mass function is:

f_{XYZ} = P(X = x; Y = y; Z = z) = \frac{n!}{x!y!z!} * p_1^xp_2^yp_3^z

Solving (b): The range of the joint pdf of X, Y and Z

Given that:

n = 20

The number of voids (x, y and z) cannot be negative and they must be integers; So:

x + y + z = n

x + y + z = 20

Hence, the range is:

\{(x,y,z)| x + y + z=20\}

Solving (c): Names and values of marginal pdf of X

We have the following parameters attributed to X:

p_1 = 5\% and n = 20

Hence, the name is: Binomial distribution

Solving (d): E(x) and Var(x)

In (c), we have:

p_1 = 5\% and n = 20

E(x) = p_1* n

E(x) = 5\% * 20

E(x) = 1

Var(x) = E(x) * (1 - p_1)

Var(x) = 1 * (1 - 5\%)

Var(x) = 1 * 0.95

Var(x) = 0.95

(e)\ P(X = 1, Y = 17, Z = 3)

In (b), we have: x + y + z = 20

However, the given values of x in this question implies that:

x + y + z = 1 + 17 + 3

x + y + z = 21

Hence:

P(X = 1, Y = 17, Z = 3) = 0

(f)\ P{X \le 1, Y = 17, Z = 3)

This question implies that:

P(X \le 1, Y = 17, Z = 3) =P(X = 0, Y = 17, Z = 3) + P(X = 1, Y = 17, Z = 3)

Because

0, 1 \le 1 --- for x

In (e), we have:

P(X = 1, Y = 17, Z = 3) = 0

So:

P(X \le 1, Y = 17, Z = 3) =P(X = 0, Y = 17, Z = 3) +0

P(X \le 1, Y = 17, Z = 3) =P(X = 0, Y = 17, Z = 3)

In (a), we have:

f_{XYZ} = P(X = x; Y = y; Z = z) = \frac{n!}{x!y!z!} * p_1^xp_2^yp_3^z

So:

P(X=0; Y=17; Z = 3) = \frac{20!}{0! * 17! * 3!} * (5\%)^0 * (85\%)^{17} * (10\%)^{3}

P(X=0; Y=17; Z = 3) = \frac{20!}{1 * 17! * 3!} * 1 * (85\%)^{17} * (10\%)^{3}

P(X=0; Y=17; Z = 3) = \frac{20!}{17! * 3!} * (85\%)^{17} * (10\%)^{3}

Expand

P(X=0; Y=17; Z = 3) = \frac{20*19*18*17!}{17! * 3*2*1} * (85\%)^{17} * (10\%)^{3}

P(X=0; Y=17; Z = 3) = \frac{20*19*18}{6} * (85\%)^{17} * (10\%)^{3}

P(X=0; Y=17; Z = 3) = 20*19*3 * (85\%)^{17} * (10\%)^{3}

Using a calculator, we have:

P(X=0; Y=17; Z = 3) = 0.07195

So:

P(X \le 1, Y = 17, Z = 3) =P(X = 0, Y = 17, Z = 3)

P(X \le 1, Y = 17, Z = 3) =0.07195

(g)\ P(X \le 1)

This implies that:

P(X \le 1) = P(X = 0) + P(X = 1)

In (c), we established that X is a binomial distribution with the following parameters:

p_1 = 5\%      n = 20

Such that:

P(X=x) = ^nC_x * p_1^x * (1 - p_1)^{n - x}

So:

P(X=0) = ^{20}C_0 * (5\%)^0 * (1 - 5\%)^{20 - 0}

P(X=0) = ^{20}C_0 * 1 * (1 - 5\%)^{20}

P(X=0) = 1 * 1 * (95\%)^{20}

P(X=0) = 0.3585

P(X=1) = ^{20}C_1 * (5\%)^1 * (1 - 5\%)^{20 - 1}

P(X=1) = 20 * (5\%)* (1 - 5\%)^{19}

P(X=1) = 0.3774

So:

P(X \le 1) = P(X = 0) + P(X = 1)

P(X \le 1) = 0.3585 + 0.3774

P(X \le 1) = 0.7359

(h)\ E(Y)

Y has the following parameters

p_2 = 85\%  and    n = 20

E(Y) = p_2 * n

E(Y) = 85\% * 20

E(Y) = 17

You might be interested in
How can I turn 20% into a decimal
NikAS [45]
0.2
had to make this longer so yea
8 0
3 years ago
Read 2 more answers
The area of a rectangular patio is 3x2 + 17x + 20 ft
krek1111 [17]

Answer:

a

Step-by-step explanation:

7 0
3 years ago
Solve for a:<br>9+2a = -3-4a​
lubasha [3.4K]

Answer:

a=-2

Step-by-step explanation:

Let me know if you need the steps tho.

5 0
3 years ago
Tell which number is prime: 18, 42, 68, 79.
fredd [130]
The number is 79 that is prime
8 0
3 years ago
Read 2 more answers
Tim is painting his storage shed who buys 4 gallons of white paint and 3 gallons of blue paint meaning of each gallon of white p
LenKa [72]

Answer:

  $128.49

Step-by-step explanation:

4 gallons at $15.72 each is 4×$15.72 = $62.88

3 gallons at $21.87 each is 3×$21.87 = $65.61

The total purchase is ...

$62.88 +65.61 = $128.49

Tim will be spending $128.49 on paint.

5 0
3 years ago
Read 2 more answers
Other questions:
  • To calculate the maximum data rate in a line we use shannon's theorem: max data rate =bandwidth x log2 (1 + s/n) what is the max
    14·1 answer
  • what is the area of a trapezoid with bases of 8.2 CM + 5.9 cm and a height of 10 cm will mark brainiest
    10·1 answer
  • In a political science class, test scores were determined to be 20 times the number of hours,h, the student studied plus 3. Whic
    8·1 answer
  • What is the average walking rate in miles per hour if the distance(miles)=0.5. And the Time (min)= 12
    5·1 answer
  • Tom has read 2 books for school. Tom read 1/2 of the first book on Thursday and 5/12 of the second book on Friday. What total fr
    5·1 answer
  • The minor arc measures of circle Z are shown in the figure use the drop down menus to complete each statement
    10·2 answers
  • Jess delivers 138 newspapers each day. Is it reasonable to estimate that jess delivers about 1,500 newspapers in 7 days? Explain
    9·1 answer
  • PLS HELP WILL GIVE BRAINLY!!
    14·1 answer
  • I will try to give brainliest just pls help
    6·1 answer
  • Is the equation a linear function or nonlinear function?
    10·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!