Gaussian Distribution, 10 pts) Let X∼N(μ,σ 2
) be a Gaussian random variable, which has the following probability density function: f(x)= exp(− 2σ 2
(x−μ) 2
)
σ 2π
1
Find E(X) and Var(X) and show all the derivation steps.
Answers
1. the expected value of X, E(X), is μ.
2. the variance of X, Var(X), is σ^2 * (-u^2 / √(2π)).
To find the expected value (E(X)) and variance (Var(X)) of a Gaussian random variable X with probability density function f(x), we need to perform the following calculations:
1. Expected Value (E(X)):
The expected value of X, denoted as E(X), is given by:
E(X) = ∫xf(x)dx
Using the given probability density function:
f(x) = (1 / √(2πσ^2)) * exp(-(x - μ)^2 / (2σ^2))
We can calculate E(X) as follows:
E(X) = ∫x * f(x) dx
= ∫x * (1 / √(2πσ^2)) * exp(-(x - μ)^2 / (2σ^2)) dx
To solve this integral, we can use the substitution method. Let's make the substitution u = (x - μ) / σ, which gives us du = dx / σ.
Substituting the values, the integral becomes:
E(X) = (1 / √(2πσ^2)) * ∫(μ + σu) * exp(-u^2 / 2) * σ du
= (1 / √(2πσ^2)) * ∫(μσ + σ^2u) * exp(-u^2 / 2) du
= (μσ / √(2πσ^2)) * ∫exp(-u^2 / 2) du + (σ^2 / √(2πσ^2)) * ∫u * exp(-u^2 / 2) du
The first integral is the integral of the standard normal distribution, which evaluates to 1:
∫exp(-u^2 / 2) du = √(2π)
The second integral is the expected value of the standard normal distribution, which is 0:
∫u * exp(-u^2 / 2) du = 0
Substituting these values back into the equation for E(X), we get:
E(X) = (μσ / √(2πσ^2)) * √(2π) + (σ^2 / √(2πσ^2)) * 0
= μ
Therefore, the expected value of X, E(X), is μ.
2. Variance (Var(X)):
The variance of X, denoted as Var(X), is given by:
Var(X) = E((X - E(X))^2)
Substituting the value of E(X) from the previous step:
Var(X) = E((X - μ)^2)
Using the given probability density function, we can calculate Var(X) as follows:
Var(X) = ∫(x - μ)^2 * f(x) dx
= ∫(x - μ)^2 * (1 / √(2πσ^2)) * exp(-(x - μ)^2 / (2σ^2)) dx
Again, we can use the substitution u = (x - μ) / σ, which gives us du = dx / σ.
Substituting the values, the integral becomes:
Var(X) = (1 / √(2πσ^2)) * ∫(σu)^2 * exp(-u^2 / 2) * σ du
= (1 / √(2πσ^2)) * ∫σ^2u^2 * exp(-u^2 / 2) * σ du
= (σ^2 / √(2πσ^2)) * ∫u^2 * exp(-u^2 / 2) du
To solve this integral, we can use integration by parts. Let's differentiate u^2 and integrate exp(-u^2 / 2):
Let f(u) = u^2, and g'(u) = exp(-u^2 / 2).
Differentiating f(u) with respect to u, we get f'(u) = 2u.
Integrating g'(u) with respect to u, we get g(u) = -√(π/2) * erf(u/√2), where erf(x) is the error function.
Using the integration by parts formula, ∫f(u) * g'(u) du = f(u) * g(u) - ∫g(u) * f'(u) du, we can evaluate the integral:
∫u^2 * exp(-u^2 / 2) du = -u^2 * √(π/2) * erf(u/√2) - ∫(-√(π/2) * erf(u/√2) * 2u du
= -u^2 * √(π/2) * erf(u/√2) + 2 * √(π/2) * ∫u * exp(-u^2 / 2) du
The remaining integral is the expected value of the standard normal distribution, which we previously calculated as 0.
Substituting the values back into the equation for Var(X), we get:
Var(X) = (σ^2 / √(2πσ^2)) * (-u^2 * √(π/2) * erf(u/√2)) + (2 * √(π/2) * 0)
= σ^2 * (-u^2 * erf(u/√2)) / √(2πσ^2)
= σ^2 * (-u^2 * erf(u/√2)) / √(2)σ
= σ^2 * (-u^2 / √2) * (erf(u/√2) / √(σ^2))
= σ^2 * (-u^2 / √2) * (erf(u/√2) / σ)
Finally, using the property that erf(x) / x approaches 2/√π as x approaches infinity, we have:
Var(X) = σ^2 * (-u^2 / √2) * (2 / √π)
= σ^2 * (-u^2 / √(2π))
Therefore, the variance of X, Var(X), is σ^2 * (-u^2 / √(2π)).
Please note that u is the standardized variable defined as (x - μ) / σ, where μ is the mean and σ is the standard deviation of the Gaussian random variable X.
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Related Questions
Seritte is saving her money to buy a new pony
which costs $675. She has already saved $25. She
earns $50 a week working at the stables. Write and
solve an equation to show how many weeks it will
take Sadie to earn enough money to buy a pony
ojo
Answers
Answer:
okay go to the quickpill app
Step-by-step explanation:
let Xk be independent and normally distributed with common mean 22 and standard deviation 11 (so their common variance is 1.1.) compute (to at least four decimal places) p(−[infinity]≤∑k=181xk≤172.89)
Answers
Let Xk be independent and normally distributed with common mean 22 and standard deviation 11 (so their common variance is 1.1.). The value of p(-[infinity] ≤ ∑k=181 Xk ≤ 172.89) ≈ 0.0265.
The common variance of independent and normally distributed random variables with a common mean can be computed with the formula
σ² = Var[Xk]
= 1.1.
This can be solved by finding the standard deviation, which is equal to the square root of the variance as
σ = sqrt(1.1)
= 1.0488.
The sum of the independent and identically distributed normal random variables can be expressed as a normal random variable with a mean equal to the sum of the individual means and a variance equal to the sum of the individual variances.
Thus, we can express this random variable as ∑k=181 Xk ~ N(22 * 181, 1.1 * 181).
To solve the given problem, we need to compute the probability that the sum of the random variables lies between -infinity and 172.89.
Since the sum of normal random variables follows the normal distribution, we can standardize the sum by subtracting the mean and dividing by the standard deviation.
Let Z be the standardized random variable.
Then Z = (X - μ)/σ
Z = (∑k=181 Xk - μ)/σ, where μ = 22 * 181 and σ = sqrt(1.1 * 181).
We need to compute P(-infinity ≤ Z ≤ (172.89 - μ)/σ) = P(Z ≤ (172.89 - μ)/σ) .
Since the standard normal distribution is symmetric about zero.
This can be solved using standard normal tables or a calculator to obtain P(Z ≤ -1.9309) ≈ 0.0265 (rounded to four decimal places).
Therefore, p(-[infinity] ≤ ∑k=181 Xk ≤ 172.89) ≈ 0.0265 (rounded to four decimal places).
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Mona has 3200 yards of fencing to enclose a rectangular area. find the dimensions of the rectangle that maximize the enclosed area. what is the maximum area?
Answers
Maximum area enclosed by Mona for the fencing is 640,000 square yards. And the dimensions( length and width ) of the rectangle to enclosed maximum area are 800 yard each.
As given in the question,
Perimeter of the rectangular area encloses for fencing = 3200 yards
Let 'x' and 'y' be the length and width of the rectangle.
2( x + y ) = 3200
⇒ x + y = 3200/2
⇒y = 1600 - x
Area of rectangular field 'A' = xy
⇒ A = x ( 1600 - x )
⇒ A = 1600x - x²
Differentiate both the side of the equation we get,
dA/dx = 1600 - 2x
For Maximum Area dA/dx = 0
1600 -2x =0
⇒ 2x = 1600
⇒ x= 800 yards
Now, y = 1600 - 800
= 800 yards
Maximum area = (800) (800)
= 640,000 square yards
Therefore, the dimensions of the rectangular area are 800yards each and its maximum area is equal to 640,000 square yards.
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Can y’all help me on question 2 please?
Answers
Answer:
35 i think is the answer but im probably wrong
Step-by-step explanation:
9) You are a contractor and charge $55 for a site visit plus an additional $25 per hour for each hour you spend
working at the site, Write and solve an equation to determine how many total hours you have to work to earn
$555 working at a site.
Okay
Answers
The contractor would have to work for 20 hours on the site to earn $555 total pay
What is total cost function?
Total cost function is the sum of the fixed charge, which is fixed cost to the person paying the contractor when the contractor visits the site and charge per hour multiplied by the number of hours spent working
TC=a+bX
TC=total cost=$555
a=fixed charged=$55
b=charge per hour=variable element of contractor's pay=$25
X=number of hours spent working=unknown
$555=$55+$25X
$555-$55=$25X
$500=$25*X
X=$500/$25
X=20
The contractor needs to work for 20 hours in order to earn a total pay of $555
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how do i solve this?
Answers
Answer:
add up ??
Step-by-step explanation:
The midpoint of AB is M(3, 1). If the coordinates of A are (1, -1), what are the
coordinates of B?
Answers
Answer:
(5,3)
Step-by-step explanation:
Find a Doctor, is a small startup that helps people find a physician that best meets their needs (location, insurance accepted, etc) During a "slow time for them, they have 9 staff members taking calls from customers. On average, one call arrives every 5 minutes (standard deviation of 5 minutes). Each staff member spends on average 18 minutes with each customer (with a standard deviation of 27 minutes) Round your answer to 2 decimal places) How long does a customer spend on average waiting on hold before they can start speaking to a representative? Minutes
Answers
On average, a customer spends approximately 1.16 minutes waiting time on hold before they can start speaking to a representative.
To find the average waiting time for a customer on hold before they can start speaking to a representative, we need to consider both the arrival rate of calls and the average service time of the staff members.
Given:
9 staff members taking calls.
On average, one call arrives every 5 minutes (standard deviation of 5 minutes).
Each staff member spends on average 18 minutes with each customer (with a standard deviation of 27 minutes).
To calculate the average waiting time, we need to use queuing theory, specifically the M/M/c queuing model. In this model:
"M" stands for Markovian or memoryless arrival and service times.
"c" represents the number of servers.
In our case, we have an M/M/9 queuing model since we have 9 staff members.
The average waiting time for a customer on hold is given by the following formula:
Waiting time = (1 / (c * (μ - λ))) * (ρ / (1 - ρ))
Where:
c = number of servers (staff members) = 9
μ = average service rate (1 / average service time)
λ = average arrival rate (1 / average interarrival time)
ρ = λ / (c * μ)
First, let's calculate the average arrival rate (λ):
λ = 1 / (average interarrival time) = 1 / 5 minutes = 0.2 calls per minute
Next, calculate the average service rate (μ):
μ = 1 / (average service time) = 1 / 18 minutes = 0.0556 customers per minute
Now, calculate ρ:
ρ = λ / (c * μ) = 0.2 / (9 * 0.0556) ≈ 0.407
Finally, calculate the waiting time:
Waiting time = (1 / (c * (μ - λ))) * (ρ / (1 - ρ))
= (1 / (9 * (0.0556 - 0.2))) * (0.407 / (1 - 0.407))
≈ 1.16 minutes
Therefore, on average, a customer spends approximately 1.16 minutes waiting on hold before they can start speaking to a representative.
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which of the following basic functions is equivalent to the piecewise-defined function f(x)= x if x≥0 −x if x<0 ? question content area bottom part 1 a. f(x)= 1 x b. f(x)=x c. f(x)=x2 d.
Answers
The basic function equivalent to the piecewise-defined function f(x) = x if x ≥ 0 and -x if x < 0 is f(x) = |x|, which represents the absolute value of x.
The given piecewise-defined function f(x) has different expressions for different intervals. For x greater than or equal to zero, f(x) takes the value of x. For x less than zero, f(x) is equal to -x. We need to find a basic function that captures this behavior.
Among the options provided, f(x) = |x| is equivalent to the given piecewise function. The absolute value function, denoted by |x|, returns the positive value of x regardless of its sign. When x is non-negative, |x| equals x, and when x is negative, |x| is equal to -x, mirroring the conditions of the piecewise-defined function.
The function f(x) = |x| represents the absolute value of x and matches the behavior of the given piecewise-defined function, making it the equivalent basic function.
In summary, the basic function equivalent to the piecewise-defined function f(x) = x if x ≥ 0 and -x if x < 0 is f(x) = |x|, which represents the absolute value of x.
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Two candles are light at the same time one burns for 6 hours and the other one lasts 2 hours. The first candles flame is 3 times longer than the other ones. How long do the both candles burn for? (please write with the way you found the answer I will give brainliest.)
Answers
At 150 minutes the length of the first candle is 3 times that of the other.
The complete question is
There are two candles of same length and same size. Both of them burn at uniform rate. The first one burns in 5 hours and the second one burns in 2 hours. Both the candles are lit together. After how many minutes the length of the first candle is 3 times that of the other?
What is an Equation ?When two algebraic expression are equated using an equal sign an equation is formed.
First candle burns for 5 hours
Second candle burns for 2 hours
Let the length of candle be l.
After t time (in minutes), length of first candle =l−(lt/300)
Length of second candle =l− (lt/120)
From the data given
l - (lt/120) = 3 * (l−(lt/300))
On simplifying
4t = 600
t = 150 minutes
Therefore after 150 minutes the length of the first candle is 3 times that of the other.
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Which describes the combined variation shown in the equation F = kxy/z?
1) F varies directly with x, and inversely with y
and z.
2)F varies directly with z, and inversely with x
and y.
3)F varies directly with y, and inversely with x and z.
4) F varies directly with x and y, and inversely
with z.
Answers
Answer:
option 4
Step-by-step explanation:
F = \(\frac{kxy}{z}\)
k is the constant of variation
• if the variables are on the numerator they vary directly
• if the variables are on the denominator they vary inversely
In this case F varies directly with x and y ( variables on numerator ) and inversely with z ( variable on denominator )
8 6/7 divided by 7?
pls help!
Answers
Answer:
\(=\frac{62}{49}\)
Step-by-step explanation:
Given: \(8\frac{6}{7} /7\)
First, convert the mixed fraction to an improper fraction:
\(8\frac{6}{7}\) = \(\frac{62}{7}\)
Then to divide, turn the division sign to a multiplication and turn 7/1 around:
\(\frac{62}{7} *\frac{1}{7}\)
= \(\frac{62}{49}\)
Answer:
62/49
Step-by-step explanation:
To divide a mixed number by a whole number, you need to convert the mixed number into an improper fraction and then divide it by the whole number. What you want to do first is convert 8 6/7 into an improper fraction by multiplying the whole number 8 by the denominator 7 and then adding the numerator 6. This gives you 62/7. Next you want to divide 62/7 by 7 by multiplying 62/7 by the reciprocal of 7, which is 1/7. This gives you (62/7) x (1/7) = 62/49. Now simplify the fraction by dividing both the numerator and denominator by their greatest common factor, which is 1. This gives you 62/49 as the simplest form.
So, 8 6/7 divided by 7 is 62/49.
Hope this helps :)
what is the x intercept x- 9y = -45
Answers
Answer:
(-45,0)
Step-by-step explanation:
Answer:
(-45, 0)
Step-by-step explanation:
A number is between 14 and 19. It has prime factors of 2 and 3. What is the number
Answers
Answer:
18
Step-by-step explanation:
18devided by 3 is 6 and 18 divided by 2 is 9
The number that has prime factors of 2 and 3 is 18.
What are prime factors?
Prime factors are factors that are prime numbers. Prime numbers are numbers that are divisible by one and itself. Examples are 2 and 3.
What are the factors of the numbers between 14 and 19?
Factors of 15 = 1, 3, 5, 15 Factors of 16 = 1, 2, 4, 8, 16Factors of 17 = 1, 17Factors of 18 = 1, 2, 3, 6, 8, 16To learn more about factors, please check: https://brainly.com/question/26073850
If x+40=? type your answer below.
Answers
Let p(n, k) be the number of ways to write the integer n as a sum of exactly k positive integers. A theorem later in the text tells us that: p(n, k)pn-1,k-1)p(n - k, k) Also, p(n,k) 0 when k n and p(n, 0) 0 and p(n,n) 1. Write a recursive algorithm to find p(n, k)
Answers
The output will be the value of p(n, k), i.e., the number of ways to write the integer n as the sum of exactly k positive integers.
The given recursive formula to calculate p(n, k) is:p(n, k) = p(n − 1, k − 1) + p(n − k, k), where p(n, k) denotes the number of ways to represent the integer n as the sum of exactly k positive integers.
Using this formula, we can generate the following recursive algorithm to compute p(n, k):
Step 1: Input the value of n and k.
Step 2: If k = 0 or k > n, then p(n, k) = 0.
Step 3: If k = n, then p(n, k) = 1.
Step 4: If k < n, then p(n, k) = p(n − 1, k − 1) + p(n − k, k).
Step 5: Return p(n, k) as the output of the algorithm.
Here's an example program in Python that implements this algorithm and returns the value of p(n, k):def partition(n, k): if k == 0 or k > n: return 0 elif k == n: return 1 else: return partition(n - 1, k - 1) + partition(n - k, k)n, k = map(int, input().split())print(partition(n, k))
The input should be provided as two space-separated integers, n and k, respectively.
Therefore, the output will be the value of p(n, k), i.e., the number of ways to write the integer n as the sum of exactly k positive integers.
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PLEASE HELP ME I BEG IMMA FAILLLLLLLLLLLLLLLL
Find the volume of a sphere with a radius of 8.6 cm. Use 3.14 for π. Round your answer to the nearest tenth.
928.9 cm3
1,497.9 cm3
2,663 cm3
665.7 cm3
Answers
Answer:
3 you welcome
Step-by-step explanation:
Answer:
2,663 cm3
Step-by-step explanation:
*WILL MARK BRAINLIEST*10 POINTS
How many triangles can be made given the following angles?
33°, 42°, 66°
A: One triangle
B: More than one triangle
C: No triangle can be made
Answers
No triangle can be made because every triangle has to be equal to 180 degrees to be considered a triangle.
Determine whether the numerical value is a parameter or a statistic. Explain your reasoning.
Sixty-two of the 97 passengers aboard the Hindenburg airship survived its explosion.
Answers
The numerical value "62" is a statistic because it is based on the sample of passengers aboard the Hindenburg airship.
In the given scenario, the numerical value is "62." We are told that there were 97 passengers aboard the Hindenburg airship and that 62 of them survived its explosion.
Since the numerical value is based on the sample (i.e., the passengers aboard the Hindenburg airship), it is a statistic. It describes the proportion of passengers who survived the explosion in the sample.
If we wanted to know the proportion of passengers who survived the explosion in the entire population (i.e., all passengers who have ever traveled on the Hindenburg airship), we would need to determine the parameter.
However, since we do not have data on the entire population, we cannot determine the parameter.
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solve for the missing side,x.
Answers
Answer:
x = 9
y = 14
Step-by-step explanation:
If we observe the two figures, the bigger one is increased by 1.5 because 12/9 = 1.5.
Smaller side = 6
Larger side = x
=> 6 x 1.5
=> x = 9
Smaller side = y
Larger side = 21
=> 21 / 1.5
=> y = 14
How much money would you need to deposit at 9% annual interest compound monthly to have K12000 in the account after 6 year
Answers
The amount of money you would need to deposit at 9% annual interest compound monthly to have K12000 in the account after 6 years is K5967.06.
To calculate the amount of money required, we can use the formula for compound interest:
Amount = Principal * (1 + (rate/n))^(n*t)
Where: Principal = the initial amount of money invested Rate = the annual interest rate (as a decimal)N = the number of times per year interest is compounded T = the number of years
The given information is as follows: Principal = PV = ?Rate = r = 9% = 0.09N = 12 (monthly compounding)T = 6 years Amount = FV = K12000
We can substitute the given values into the formula to solve for
\(PV.K12000 = PV * (1 + (0.09/12))^ (12*6)K12000 = PV * (1.0075)^72PV = K12000/(1.0075)^72PV\)
\(= K12000/2.0128PV\)
\(= K5967.06\)
Therefore, the amount of money you would need to deposit at 9% annual interest compound monthly to have K12000 in the account after 6 years is K5967.06.
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If an item is discounted by 5 percent, and the final price is 57, what is the original price of the item?
Answers
Answer:
$59. 85 or rounding, $60 (I AM NOT COMPLETELY SURE)
Step-by-step explanation
I changed the percentage into a decimal (.05) and then multiplied it by the final price (57). After multiplying, I got, $2.85. I then added that to the final price to get, $59.85.
Write the expression as an equivalent expression in the form 2". Simplify your answer as much as possible, and enter your answer as a fraction. VI = x" for n = help (fractions
Answers
The equivalent expression in the form 2 is 40.
We are to write the expression VI = x in an equivalent expression in the form 2. To write the expression VI = x as an equivalent expression in the form 2, we should express the given expression VI = x in terms of exponential form.
The exponential form of VI is:
VI = 2³ * 5¹
So, the expression VI = x can be written as:
2³ * 5¹ = x
2³ * 5 = x
So, the equivalent expression in the form 2 is 2³ * 5.
Simplifying further,
2³ * 5 = 8 * 5 = 40
Hence, the expression VI = x in the form 2 is 40.
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Consider the statement "The square of any odd integer is odd." (a) Rewrite the statement in the form Forall n, . (Do not use the words "if or "then.") (b) Rewrite the statement in the form Forall n, if then. (Make sure you use the variable n when you fill in each of the second two blanks.) (c) Write a negation for the statement.
Answers
(a) For all n, if n is an odd integer, then the square of n is odd. (b) For all n, if n is an odd integer, then the square of n is an odd integer. (c) There exists an integer n such that the square of n is not odd.
(a) For all n, if n is an odd integer, then the square of n is odd .The statement is rewritten in the universal form "For all n" indicating that the property holds true for every odd integer. (b) For all n, if n is an odd integer, then the square of n is an odd integer. The statement is rewritten in the universal form with an implication "if...then" structure, specifying that if n is an odd integer, then the square of n will also be an odd integer. (c) There exists an integer n such that the square of n is not odd. The negation of the statement states that there exists at least one integer n for which the square of n is not an odd integer, implying that there is a counterexample to the original statement.
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does anyone know this? URGENT!!
Answers
Answer: Y=3, X=-2
Step-by-step explanation:
Answer:
\(y = \frac{3}{2} x + 3\)
Step-by-step explanation:
slope is a positive 3/2
y intercept is 3
In ΔVWX, x = 41 cm, v = 17 cm and ∠W=134°. Find the length of w, to the nearest centimeter.
Answers
Answer:
54
Step-by-step explanation:
deltamath gave me the answer
The length of w is 54 cm.
What is Trigonometry?One of the six mathematical functions used to express the side ratios of right triangles, the trigonometric function includes the sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). The figure shows these six trigonometric functions in respect to a right triangle. Surveying, engineering, and navigation issues where one of a right triangle's acute angles and the length of a side are known but the lengths of the other sides need to be determined can be solved with ease using trigonometry.
As per the given data:
x = 41 cm
v = 17 cm
∠W = 134°
To find the length of w:
From the cosine law: \(cos(W) = \frac{x^2 + v^2 - w^2}{2xv}\)
\(w = \sqrt{x^2 + v^2 - 2xvcos(W)}\\w = \sqrt{(41)^2 + (17)^2 - 2(41)(17)cos(W)}\\w = \sqrt{1681 + 289 + 968.35} = \sqrt{2938.35}\\w = 54 \:cm\)
Hence, the length of w is 54 cm.
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Which expression's product is modeled by the tiles?
Answers
Answer:
A
the tiles add up to x^2 + 4x + 3
(x + 1) (x + 3)
factor it out, and you get x^2 + 4x + 3.
What does each variable in the Slope-Intercept Form equation represent?
Answers
Ashley had 4/ 5 of a spool of yarn. She used 2/5 of it for her project. What fraction of the spool was used for her project? Write your answer in simplest form
Answers
Ashley used 8/25 of the spool for her project.
To determine the fraction of the spool that Ashley used for her project, we need to multiply the fraction of the spool she had (4/5) by the fraction she used (2/5):
(4/5) * (2/5) = 8/25
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Please help me out. Will make as brainiest!
Answers
Answer:
Quadratic function
Step-by-step explanation: