(i) For X ~ Laplace(0,1):
E(X) = 0, Var(X) = 2.
(ii) If X ~ Laplace(0,1) and Y = bX + μ:
Y ~ Laplace(μ, b).
(iii) For W ~ Laplace(2,8):
E(W) can be approximated numerically.
Var(W) = 128.
(i) If X ~ Laplace(0,1), we need to find the expected value (E(X)) and variance (Var(X)).
The Laplace(0,1) distribution has μ = 0 and b = 1. Substituting these values into the PDF, we have:
fX(x) = (1/2) * exp(-|x|)
To find E(X), we integrate x * fX(x) over the entire range of X:
E(X) = ∫x * fX(x) dx = ∫x * [(1/2) * exp(-|x|)] dx
Since the Laplace distribution is symmetric about the mean (μ = 0), the integral of an odd function over a symmetric range is zero. Therefore, E(X) = 0 for X ~ Laplace(0,1).
To find Var(X), we use the formula:
Var(X) = E(X^2) - [E(X)]^2
First, let's find E(X^2):
E(X^2) = ∫x^2 * fX(x) dx = ∫x^2 * [(1/2) * exp(-|x|)] dx
Using the symmetry of the Laplace distribution, we can simplify the integral:
E(X^2) = 2 * ∫x^2 * [(1/2) * exp(-x)] dx (integral from 0 to ∞)
Solving this integral, we get:
E(X^2) = 2
Now, substitute the values into the variance formula:
Var(X) = E(X^2) - [E(X)]^2 = 2 - 0 = 2
Therefore, for X ~ Laplace(0,1), E(X) = 0 and Var(X) = 2.
(ii) To show that Y = bX + μ follows a Laplace(μ, b) distribution, we need to find the probability density function (PDF) of Y.
Using the transformation method, let's express X in terms of Y:
X = (Y - μ)/b
Now, calculate the derivative of X with respect to Y:
dX/dY = 1/b
The absolute value of the derivative is |dX/dY| = 1/b.
To find the PDF of Y, substitute the expression for X and the derivative into the Laplace(0,1) PDF:
fY(y) = fX((y-μ)/b) * |dX/dY| = (1/2) * exp(-|(y-μ)/b|) * (1/b)
Simplifying this expression, we get:
fY(y) = 1/(2b) * exp(-|y-μ|/b)
This is the PDF of a Laplace(μ, b) distribution, thus showing that Y ~ Laplace(μ, b).
(iii) For W ~ Laplace(2,8), we need to find E(W) and Var(W).
The PDF of W is given by:
fW(w) = (1/16) * exp(-|w-2|/8)
To find E(W), we integrate w * fW(w) over the entire range of W:
E(W) = ∫w * fW(w) dw = ∫w * [(1/16) * exp(-|w-2|/8)] dw
This integral can be challenging to solve analytically. However, we can approximate the expected value using numerical methods or software.
To find Var(W), we can use the property that the variance of the Laplace distribution is given by 2b^2, where b is the scale parameter.
Var(W) = 2 * b^2
= 2 * (8^2)
= 2 * 64
= 128
Therefore, Var(W) = 128 for W ~ Laplace(2,8).
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With relevant examples on Free un-damped vibrations and Free damped vibrations, describe how Modeling with Higher Order Differential Equations is done.
Modeling with higher order differential equations is a powerful tool used to describe the behavior of systems in free un-damped and free damped vibrations.
By considering the forces acting on the system and applying Newton's laws, we can derive higher order differential equations that capture the dynamics of the system. When studying free un-damped vibrations, we consider systems that oscillate without any external damping or resistance. One common example is a mass-spring system, where a mass is attached to a spring and allowed to oscillate freely.
By applying Newton's second law and considering the forces acting on the mass (including the spring force and the inertia of the mass), we can derive a second-order differential equation, typically of the form mx''(t) + kx(t) = 0, where x(t) represents the displacement of the mass as a function of time, m is the mass of the object, and k is the spring constant. Solving this equation provides information about the system's natural frequency and the amplitude of the oscillations.
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Graph the function f(x) = 2x. Which features are correctly stated?
A) x-intercept: none
B) y-intercept: (0, 2)
C) asymptote: x = 0
D) as x → ∞, f(x) → ∞
E) as x → −∞, f(x) → 0
Answer:
a, d, e
Step-by-step explanation:
on usatestprep
The features of the function are given as follows:
Horizontal asymptote: y = 5.
Vertical asymptote: x = 2.
x-intercept: No x-intercept.
y-intercept: (0,-5).
Hole: No holes.
What are the features of the function?The horizontal asymptote is the limit of f(x) as x goes to infinity. To obtain this limit, we consider only the terms with the highest exponent of the numerator and of the denominator, hence:
here, we have,
g(x) = -5x/x -> y = 5 is the horizontal asymptote.
The vertical asymptote is the value of x for which the function is not defined, hence it is the zero of the denominator, thus:
x - 2 = 0 -> x = 2.
The x-intercept is the point (x,0), in which x is found when y = 0, hence:
-5x + 10 = 0
-5x = -10
5x = 10
x = 2.
However, the function is not defined at x = 2, meaning that it has no x-intercept.
The y-intercept is the value of y when x = 0, hence:
y = 10/-2 = -5.
The coordinates are (0,-5).
The entire function can be simplified as follows:
(-5x + 10)/(x - 2) = [-5(x - 2)]/(x - 2) = -5.
Meaning that the function has no holes.
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complete question:
Determine each feature of the graph of the given function.
f(x) = -5+10
2
Horizontal Asymptote: y =
Vertical Asymptote: x =
x-Intercept:
y-Intercept: (0,
Hole: (
,0)
No horizontal asymptote
No vertical asymptote
No x-intercept
No y-intercept
No hole
reaction 1: y x- → y- x reaction 2: y z- → y- z reaction 3: z x- → z- x the three substances in order of increasing oxidizing ability (strength as an oxidizing agent). x,y,z
The substances in order of increasing oxidizing ability are: x < z < y.
To determine the order of increasing oxidizing ability of substances x, y, and z, we need to analyze the given reactions.
In reaction 1, y oxidizes x by removing an electron from x, forming y- and x+. This suggests that y has a higher oxidizing ability than x since it can accept an electron.
In reaction 2, y oxidizes z by removing an electron from z, forming y- and z+. Similar to reaction 1, y acts as an oxidizing agent in this reaction as well.
In reaction 3, z oxidizes x by removing an electron from x, forming z- and x+. Here, z exhibits oxidizing behavior by accepting an electron.
Based on the reactions, we can conclude that y has the highest oxidizing ability since it is involved in both reaction 1 and reaction 2 as an oxidizing agent. Z comes next in terms of oxidizing ability as it participates in reaction 3. Finally, x appears to have the lowest oxidizing ability as it is being oxidized in both reaction 1 and reaction 3. Therefore, the substances in order of increasing oxidizing ability are: x < z < y.
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OOOF MATH HELP EASY thank you
Answer: B.)
Step-by-step explanation:
Answer: B
Step-by-step explanation:
what is the slope and y intercept of the line with equation 3x-6y=7
Answer:
x intercept = (7/3, 0)
y intercept = (0, -7/6)
slope = 1/2
Step-by-step explanation:
Please Help! Draw the graph of an absolute value function that has these key features:
vertex: (2,-3)
increasing: (2, ∞)
decreasing: (-∞, 2)
domain: all real numbers
range: [-3, ∞)
end behavior: As x approaches negative infinity, f(x) approaches positive infinity. As x approaches positive infinity, f(x) approaches positive infinity.
Answer:edmentum
Step-by-step explanation:
A teacher is correcting a student's homework
assignment, but she cannot read one line of the
student's work. The student's work is shown
below, with a blank space representing the
missing line.
Step 1:
- 2x + 11 = -5
Step 2: -2x + 11 – 11 = -5 – 11
Step 3:
-2x = -16
Step 4:
Step 5:
1x = 8
Step 6:
x= 8
Which of the following represents the correct
Step 4 and the property that would be used to
justify the step?
Answer without Explanation:
B
A genetic experiment with peas resulted in one sample of oping that consisted of 443 green peas and 10 yelow peas a. Construct a 50% confidence intervallo state of the percentage of b. Based on the confidence interval do the results of the experiment appear to control the expectation that 20% of the ring peas won Construct a son contence terva Express the percentages in conform
a) Therefore, the 50% confidence interval for the percentage of yellow peas in the population is approximately (0.47%, 3.93%).
b) Since the confidence interval does not contain the value of 20%, we can say that the results of the experiment do not support the expectation that 20% of the peas should be yellow.
c) The 90% confidence interval for the percentage of green peas in the population is approximately (95.87%, 99.53%).
The data given in the problem is:
Sample size (n) = 443 + 10 = 453
Number of yellow peas (x) = 10
Number of green peas (n-x) = 443
Since the sample size is very large (n > 30), we can use the normal distribution to find the confidence interval.
a) To construct a 50% confidence interval for the percentage of yellow peas in the population, we use the following formula:
Lower limit = p - z(α/2)√(p(1-p)/n)
Upper limit = p + z(α/2)√(p(1-p)/n)
where: p = x/n = 10/453 = 0.022 (proportion of yellow peas in the sample)
z(α/2) = z(0.25) = 0.674 (z-value for a 50% confidence level)
Plugging in the values, we get:
Lower limit = 0.022 - 0.674√(0.022(1-0.022)/453) ≈ 0.0047
Upper limit = 0.022 + 0.674√(0.022(1-0.022)/453) ≈ 0.0393
Therefore, the 50% confidence interval for the percentage of yellow peas in the population is approximately (0.47%, 3.93%).
b) Since the confidence interval does not contain the value of 20%, we can say that the results of the experiment do not support the expectation that 20% of the peas should be yellow.
c) To construct a 90% confidence interval for the percentage of green peas in the population, we can use the same formula as before with x = 443:
Lower limit = p - z(α/2)√(p(1-p)/n)
Upper limit = p + z(α/2)√(p(1-p)/n)where:
p = x/n = 443/453 = 0.977 (proportion of green peas in the sample)
z(α/2) = z(0.05) = 1.645 (z-value for a 90% confidence level)
Plugging in the values, we get:
Lower limit = 0.977 - 1.645√(0.977(1-0.977)/453) ≈ 0.9587
Upper limit = 0.977 + 1.645√(0.977(1-0.977)/453) ≈ 0.9953
Therefore, the 90% confidence interval for the percentage of green peas in the population is approximately (95.87%, 99.53%).
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(a) y = 5z + x simplify x
Answer:
x=y−5z
Step-by-step explanation:
y=5z+x
Step 1: Flip the equation.
x+5z=y
Step 2: Add -5z to both sides.
x+5z+−5z=y+−5z
x=y−5z
Answer:
x=y−5z
plz mark me as brainliest
Answer:
x = ay - 5z
Step-by-step explanation:
Solve for x by simplifying both sides of the equation, then isolating the variable.
match the range of the function f(x)=x^2+2x-1 to its domain
2
-2
3
-3
Answer:
See solutions below
Step-by-step explanation:
Match the range of the function f(x)=x^2+2x-1 to its domain
Domains of the function are the input values x i.e 2, -2, 3 and 3
corresponding f(x) for each values are the range
when x = 2
f(2)=2^2+2(2)-1
f(2) = 4 + 4 - 1
f(2) = 7
when x = -2
f(-2)=(-2)^2+2(-2)-1
f(-2) = 4 - 4 - 1
f(-2) = -1
when x = 3
f(3)=3^2+2(3)-1
f(3) = 9 + 6 - 1
f(3) = 14
when x = -3
f(-3)=(-3)^2+2(-3)-1
f(-3) = 9 -6 - 1
f(-3) = 2
Will mark brainliest if correct.
Answer:
b
Step-by-step explanation:
Answer:
i am pretty sure the answer is d
Step-by-step explanation:
let me know if this is wrong
The student council had a cupcake fundraiser. The cupcakes were donated, but the icing, plates and candy were purchased for $50. The
student council sold the cupcakes for $1.50 each.
Which equation shows the relationship between the profit, p, and the number of cupcakes sold, n?
the function \:h=-0.01x^2+0.9x models the height in feet, h, of a soccer ball as it travels a horizontal distance in feet, x. What is the maximin height of the soccer ball and how long does it take for the ball to reach this height
Answer:
Nice!
Step-by-step explanation:
Select the equivalent expression.
(64.2)^3=?
Choose 1 answer:
A )
6^64x2^3
B )
6^12x6
C )
6^64x6
D )
6^12x2^3
(6^4 . 2)³ = 6^12 . 2³
The answer is: D
write two inequalities to
compare -12 and -6
write two inequalities to compare 8 and -10
Answer:
First one: -12 < -6
-6 > -12
Second one:
-8 > -10
-10 < -8
Step-by-step explanation:
Well you just compare them with greater then or less then signs.
Melissa and Matt both started a card collection at the same time. Melissa started her card collection with 100 cards and added 13 cards to her collection each week. Matt started his card collection with 130 cards and added 8 cards to his collection each week. After how many weeks did Melissa and Matt have the same number of cards in their collections?
Answer:
a few weeks
Step-by-step explanation:
What’s the answer???
Answer:
$50, 100
Step-by-step explanation:
dr. ellison says that the equation y = -3x 7 has a solution of (2, 13). is dr. ellison right or wrong
Dr. Ellison is wrong.
To determine if the equation y = -3x + 7 has a solution of (2, 13), we can substitute the values of x and y into the equation and check if it holds true.
Substituting x = 2 and y = 13 into the equation:
13 = -3(2) + 7
13 = -6 + 7
13 = 1
The equation does not hold true since 13 is not equal to 1.
Therefore, (2, 13) is not a solution to the equation y = -3x + 7.
Hence, Dr. Ellison is incorrect in stating that (2, 13) is a solution to the equation.
An equation is a mathematical statement that indicates that two expressions are equal. It consists of variables, constants, and mathematical operations such as addition, subtraction, multiplication, division, exponentiation, etc. Equations are used to represent relationships between quantities and to solve problems in various branches of mathematics and science.
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What would u do to find the missing value plz helpp ASAP plz I’ll give brainliest!
Answer:
23 [tex]\frac{1}{24}[/tex]
Step-by-step explanation:
you subtract 82 [tex]\frac{2}{3}[/tex] by 59 [tex]\frac{5}{8}[/tex] and you'll get the answer
3 of the 100 digital video recorders (DVRs) in an inventory are known to be defective. What is the probability you randomly select an item that is not defective?
Answer:
97/100
Step-by-step explanation:
Probability calculates the likelihood of an event occurring. The likelihood of the event occurring lies between 0 and 1. It is zero if the event does not occur and 1 if the event occurs.
For example, the probability that it would rain on Friday is between o and 1. If it rains, a value of one ids attached to the event. If it doesn't a value of zero is attached to the event.
Probability that the DVRs is not faulty = total number of faulty videos / total number of videos
total number of faulty videos = 100 - 3 = 97
total number of videos = 100
97/100
Sergio ate 3.5 cookies. Each cookie contained 5.7 grams of sugar. How many grams of sugar did Sergio eat?
Answer:
19.95
Step-by-step explanation:
5.7 times 3 = 17.1
5.7 ÷ 2 = 2.85
17.1 + 2.85 = 19.95
Please help I can’t understand
My brain is gonna explode
No joke I’m serious
I search it but I failed sorry
for not answering I just don't know
Determine whether the claim stated below represents the null hypothesis or the alternative hypothesis. If a hypothesis test is performed, how should you interpret a decision that (a) rejects the null hypothesis or (b) fails to reject the null hypothesis? A scientist claims that the mean incubation period for the eggs of a species of bird is at least 31 days. Does the claim represent the null hypothesis or the alternative hypothesis? Since the claim (a) How should ! a statement of equality, it represents the ?|
The claim represents the alternative hypothesis, and if the null hypothesis is rejected, it implies evidence for the alternative hypothesis.
The claim "the mean incubation period for the eggs of a species of bird is at least 31 days" represents the alternative hypothesis.
In hypothesis testing, the null hypothesis (H0) is the statement that is assumed to be true or the statement of no effect or no difference. The alternative hypothesis (Ha) is the statement that contradicts or opposes the null hypothesis and represents the possibility of an effect or a difference.
If a hypothesis test is performed and it rejects the null hypothesis (H0), it means that there is sufficient evidence to support the alternative hypothesis (Ha). In the context of the given claim, if the null hypothesis is rejected, it would imply that there is evidence to suggest that the mean incubation period for the eggs of the species of bird is less than 31 days.
On the other hand, if the hypothesis test fails to reject the null hypothesis (H0), it means that there is not enough evidence to support the alternative hypothesis (Ha). In the given claim, if the null hypothesis is not rejected, it would imply that there is not enough evidence to conclude that the mean incubation period for the eggs of the species of bird is less than 31 days.
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Find the area of the surface.
The part of the surface
z = xy
that lies within the cylinder
x² + y² = 64.
The area of the surface is 2π(3√3 - 1). The surface area integral is given by A = ∬D √(1 + (dz/dx)² + (dz/dy)²) dA.
To find the area of the surface that lies within the cylinder x² + y² = 64 and is defined by z = xy, we can use the surface area integral.
The surface area integral is given by:
A = ∬D √(1 + (dz/dx)² + (dz/dy)²) dA
where D represents the region on the xy-plane that corresponds to the surface.
In this case, the region D is the circle with radius 8 (which is obtained by solving x² + y² = 64). To evaluate the integral, we need to determine the partial derivatives dz/dx and dz/dy.
Taking the partial derivative of z with respect to x:
∂z/∂x = y
Taking the partial derivative of z with respect to y:
∂z/∂y = x
Substituting these values into the surface area integral formula:
A = ∬D √(1 + y² + x²) dA
Since D is a circle with radius 8, we can express the integral in polar coordinates. The limits of integration for r are 0 to 8, and the limits of integration for θ are 0 to 2π.
A = ∫₀²π ∫₀⁸ √(1 + r²sin²θ + r²cos²θ) r dr dθ
Simplifying the expression under the square root:
A = ∫₀²π ∫₀⁸ √(1 + r²) r dr dθ
Now we can evaluate the integral:
A = ∫₀²π [(1/3)(1 + r²)^(3/2)]⁸₀ dθ
A = ∫₀²π (1/3)(9√3 - 1) dθ
A = (1/3)(9√3 - 1) ∫₀²π dθ
A = (1/3)(9√3 - 1)(2π - 0)
A = 2π(3√3 - 1)
The area of the surface is 2π(3√3 - 1).
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I need help ASAP
How do I find the radius of a cone even if they don’t give the diameter
Who ever can answer this gets 30 points!!!
Answer:
dowload the link up the other one answer
Let W = {(0,x, y, z); x – 3y + 92 = 0} be a subspace of R*. Then a basis for W is: {(0,6.1,0), (0,-9,0,1)) None of the mentioned {(0,-6,1,0), (0,9,0,1)) {(0,3,1,0), (0,-9,0,1)) -4 12 -1 -1 Let k be a real number and A = k 2 lo 7 1 Then A is a singular matrix if None of the mentioned k=15/2 k=10 k=5
A basis for W is {(0,-6,1,0), (0,9,0,1)}.
Matrix A is a singular matrix when k = 1.
To find the basis for the subspace W and determine the value of k for matrix A, let's go through each part separately.
Part 1: Basis for W
The subspace W is defined by the equation x – 3y + 92 = 0. In order to find a basis for W, we need to find two linearly independent vectors that satisfy this equation.
From the given options, the vectors {(0,6.1,0)} and {(0,-9,0,1)} do not satisfy the equation x – 3y + 92 = 0.
However, the vectors {(0,-6,1,0)} and {(0,9,0,1)} satisfy the equation x – 3y + 92 = 0. Therefore, a basis for W is {(0,-6,1,0), (0,9,0,1)}.
Part 2: Singular Matrix A
Matrix A = [k 2 10 7 1].
To determine if A is a singular matrix, we need to check if its determinant is equal to zero.
Calculating the determinant of A:
det(A) = k(21 - 710) - 2(27 - 101) + 10(27 - 101)
= k(-68) - 2(-14) + 10(4)
= -68k + 28 + 40
= -68k + 68
A is a singular matrix if det(A) = -68k + 68 = 0.
Solving the equation -68k + 68 = 0, we find k = 1.
Therefore, A is a singular matrix when k = 1.
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The drawing below shows the dimensions of Kim kite what is the area of Kim's Kite
Answer:
B 48 in.
Step-by-step explanation:
the formula is a=bh/2 and that formula will get you 48 in^2.
A box contains three cards. On one card there is a question mark (Upper Q), on another card there is a pear (Upper P), and on the third card there is a moon (Upper M). Two cards are to be selected at random with replacement. Complete parts (a) through (e) below. a) Determine the number of sample points in the sample space.
Complete question :
Determine the probability that two pears are selected The probability is(Simplity your answer.)
d) Determine the probability that a card containing a PEAR and then a card containing a question mark are selected
Answer:
(QQ, QP, QM, QP, PP, MP, QM, PM, MM) ;
Step-by-step explanation:
Given :
Marking on card :
Question mark = Q
Pear = P
Moon = M
Selecting two cards with replacement :
Sample space :
_____ Q ____ P _____ M
Q ___QQ ___QP ___ QM
P ___QP ____PP ___ MP
M__ QM ____PM ___MM
(QQ, QP, QM, QP, PP, MP, QM, PM, MM) ;
1/9 ;
2/9
P(selecting 2 pears) = (PP)
P(p) = required outcome / Total possible outcomes = 1 / 9
Card containing a pear and then a question mark ;
(PQ or QP).
P(p) = 1/3 ; P(Q) = 1/3
P(PQ) + P(QP)
(1/3 * 1/3) + (/3 * 1/3)
1/9 + 1/9
(1 + 1) / 9
= 2/9
eight thousand eight and eight tenths in standard form help plz
Answer:
8,088.8
Step-by-step explanation:
In a population,weights of females are normally distributed with mean 52kg and standard deviation 6kg. Weights of males are normally distributed with mean 75kg and standard deviation 8kg. One male and one female are chosen at random. (a) What is the probability that the male is heavier than 81kg? [3marks] (b) What is the probability that the female is heavier than the male? Hint: If X and Y are independent Normal random variables then, for every ab e R,aX+bY has a Normal distribution. [3marks] (c If the male is above average weight(75kg),what is the probability that he is heav than 81kg?
The probability that he weighs more than 81 kg given that he is above the average weight is 0.4532.
(a) Probability that male is heavier than 81 kg can be found using the z-score formula;
z = (x - μ) / σ = (81 - 75) / 8 = 0.75P(z > 0.75) = 1 - P(z < 0.75)
Now referring to z-tables, we find that P(z < 0.75) = 0.7734
Therefore, P(z > 0.75) = 1 - P(z < 0.75) = 1 - 0.7734 = 0.2266
Thus, the probability that the male is heavier than 81 kg is 0.2266.
(b) Let X be the weight of female and Y be the weight of male. We know that X ~ N(52, 6) and Y ~ N(75, 8)
Let Z = X - Y then, Z ~ N(52 - 75, sqrt(6^2 + 8^2)) = N(-23, 10)
Now, we need to find the probability that X > Y. i.e P(X - Y > 0)P(X - Y > 0) can be written as P(Z > -23/10)
Now, referring to the z-tables, we can find that P(Z > -2.3) = 0.9893
Therefore, P(X > Y) = P(X - Y > 0) = 0.9893.
(c) Given that male weight is above average weight, we need to find the probability that he weighs more than 81 kg i.e. P(Y > 81 | Y > 75)
This is conditional probability and can be found using Bayes' Theorem:
P(Y > 81 | Y > 75) = P(Y > 75 | Y > 81) * P(Y > 81) / P(Y > 75)
Now, P(Y > 75 | Y > 81) = 1, P(Y > 81) can be found as:
P(Y > 81) = P(Z > (81 - 75) / 8) = P(Z > 0.75) = 1 - P(Z < 0.75) = 1 - 0.7734 = 0.2266
Also,P(Y > 75) = P(Z > 0) = 0.5Therefore,P(Y > 81 | Y > 75) = 1 * 0.2266 / 0.5 = 0.4532
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The given information is that weights of females are normally distributed with mean 52kg and standard deviation 6kg. Weights of males are normally distributed with mean 75kg and standard deviation 8kg. One male and one female are chosen at random.
The probability that the male is heavier than 81kg is 0.2266.
The probability that the female is heavier than the male is 0.0107.
If the male is above average weight(75kg),then the probability that he is heavy than 81kg is 0.2266.
a) The weight of males is normally distributed with a mean of 75kg and a standard deviation of 8kg. The probability that the male is heavier than 81kg can be calculated as follows: z = (81 - 75) / 8
= 0.75P(Z > 0.75)
= 0.2266
Therefore, the probability that the male is heavier than 81 kg is 0.2266.
b) If X and Y are independent Normal random variables, then, for every ab e R, aX + bY has a Normal distribution. The weights of males and females are independent, normally distributed random variables with means of 75kg and 52kg and standard deviations of 8kg and 6kg respectively. Therefore, the difference in weights of the male and the female is a normally distributed random variable with mean μ = 75 - 52 = 23kg and standard deviation σ = √(8² + 6²) = √(100) = 10kg. Let Z be a standard normal variable. The probability that the female is heavier than the male can be calculated as follows:
P(X < Y) = P(X - Y < 0) = P[(X - Y - μ)/σ < (-23)/10] = P(Z < -2.3) = 0.0107
Therefore, the probability that the female is heavier than the male is 0.0107.
c) If the male is above average weight, then we can assume that he weighs 75 kg. The probability that he is heavier than 81 kg can be calculated as follows:
z = (81 - 75) / 8 = 0.75P(Z > 0.75) = 0.2266
Therefore, the probability that the male is above average weight and heavier than 81 kg is 0.2266.
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