Let X 1
,X 2
,…,X n
be a sequence of independent exponential random variables with rate parameters λ 1
,λ 2
,…,λ n
, respectively. (a) Prove that the random variable M=min(X 1
,X 2
,…,X n
) is an exponential random variable with rate parameter λ=∑ i=1
n
λ i
. (b) Prove that the probability that the minimum M is realized by the random variable X j
is ∑ i=1
n
λ i
λ j
Answers
The problem asks to prove two statements related to the minimum of a sequence of independent exponential random variables. Firstly, it needs to be shown that the minimum, denoted as M, is itself an exponential random variable with a rate parameter equal to the sum of the individual rate parameters. Secondly, it needs to be proven that the probability of the minimum being realized by a specific random variable, Xj, is equal to the ratio of the individual rate parameter of that variable to the sum of all the rate parameters.
:
(a) To prove that M is an exponential random variable with rate parameter λ=∑i=1nλi, we need to show that it follows the properties of an exponential distribution. Since X1, X2, ..., Xn are independent exponential random variables, their minimum M can be written as M=min(X1, X2, ..., Xn). The probability density function (PDF) of M can be derived using the order statistics. From the PDF, it can be shown that M follows an exponential distribution with rate parameter λ=∑i=1nλi.
(b) To prove that the probability of the minimum M being realized by Xj is ∑i=1nλiλj, we need to show that the ratio of the rate parameter of Xj to the sum of all the rate parameters gives the desired probability. This can be done by considering the complementary event of M being realized by Xj and evaluating its probability using the properties of exponential random variables. The result will be ∑i=1nλiλj, indicating the probability of M being realized by Xj.
Both proofs involve applying the properties of exponential random variables and manipulating the PDFs and probabilities to arrive at the desired results.
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Related Questions
One factor of f (x ) = 4 x cubed minus 4 x squared minus 16 x + 16 is (x – 2). What are all the roots of the function? Use the Remainder Theorem.
Answers
\(f(x) =4 x {}^{3} - 4x {}^{2} - 16x + 16\)
\(divide \: by \: x - 2\)
\(4x {}^{2} \: into \: (x - 2) = 4x {}^{3} - 8x {}^{2} \)
\(g(x) = 4x {}^{2} - 16x + 16\)
\(4x \: into \: (x - 2) = 4x {}^{2} - 8x\)
\(z(x) = - 8x + 16\)
\( - 8 \: into \: (x - 2) = - 8x + 16 \\ no \: remainder\)
\(f(x) = (x - 2)(4x {}^{2} + 4x - 8)\)
\(f(x) = 0 \\ x - 2 = 0 \: \: \: \: \: \: \: \: \: 4x {}^{2} + 4x - 8 = 0 \\ \)
\(x = \frac{ - 4 + \sqrt{4 {}^{2} - 4(4)( - 8) } }{2(4)} = \frac{ - 4 + \sqrt{144} }{8} = \frac{ - 4 + 12}{8} = 1\)
\(x = 2\)
\(x = \frac{ - 4 - 12}{8} = \frac{ - 16}{8} = - 2\)
Answer: B
Step-by-step explanation:
Edge 2023
Tommy was going to sell all of his Pokemon
cards to buy NBA 2K21. After selling half of
them he changed his mind and bought 17
more Pokemon cards. How many did he start
with if he now has 34?
Answers
Answer: \(x = 34\)
Step-by-step explanation:
Let x be the original amount of Pokémon cards Tommy had, we know that he sold half of them; this can be express as:
\(\frac{x}{2}\)
After this he changed his mind and bought 17 cards, then he has:
\(\frac{x}{2} + 17\\\)
We know that he has 34, then we have the equation:
\(\frac{x}{2} + 17 = 34\)
Solving for x we have:
\(\frac{x}{2} + 17 = 34\)
\(\frac{x}{2} = 34 - 17\)
\(\frac{x}{2} = 17\)
\(x = (2)(17)\)
\(x = 34\)
A die is rolled and a coin is flipped simultaneously. the number rolled on the die and whether the coin lands heads or tails is recorded. how many outcomes are in the sample space? 8 6 10 12
Answers
Answer: 12
Step-by-step explanation:
Prove the property a × b = − b × a of this theorem.
Let a= (a1, a2, a3) and b= (b1, b2, b3)
Then,
a × b = xxxxx
= (−1) * xxxx
= − b × a.
Fill in for the italic x's
Answers
The property states that when two vectors a and b are multiplied together, the result is equal to the negative of b multiplied by a. This can be shown by multiplying the components of both vectors and then multiplying the result by -1.
a×b = (a2*b3 - a3*b2, a3*b1 - a1*b3, a1*b2 - a2*b1)
= (a2*b3 - a3*b2, a3*b1 - a1*b3, a1*b2 - a2*b1)
= (−1) * (a2*b3 - a3*b2, a3*b1 - a1*b3, a1*b2 - a2*b1)
= (−1) * (b2*a3 - b3*a2, b3*a1 - b1*a3, b1*a2 - b2*a1)
= − b × a.
The property a × b = − b × a states that when two vectors a and b are multiplied, the result is equal to the negative of b multiplied by a. This can be proven by first expanding the components of both vectors. For a vector a=(a1,a2,a3) and b=(b1,b2,b3), the expanded form of a×b is (a2*b3 - a3*b2, a3*b1 - a1*b3, a1*b2 - a2*b1). This can then be multiplied by -1 to get the expanded form of -b×a, which is (b2*a3 - b3*a2, b3*a1 - b1*a3, b1*a2 - b2*a1). Since this is equal to the original form of a×b, this proves the property.
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If the first three terms of the expansion of (1 + px)^n in ascending powers of x are 1+20x+160x, find the values of n and p.
Answers
The values of n and p are n=3 and p=20/3
How to find the values of n and p?We are given that the first three terms of the expansion of \((1+px)^n\) in ascending powers of x are \(1+20x+160x^2\). We need to find the values of n and p.
We know that the general formula for the binomial expansion of (1+px)^n is:
\((1+px)^n = C(n,0) + C(n,1)(px)^1 + C(n,2)(px)^2 + ... + C(n,r)(px)^r + ...\)
where C(n,r) is the binomial coefficient which is given by:
C(n,r) = n! / (r! * (n-r)!)
where n! is the factorial of n.
Since the first three terms of the expansion are given to us, we can write:
C(n,0) = 1
\(C(n,1)(px)^1 = 20x\)
\(C(n,2)(px)^2 = 160x^2\)
From the first equation, we get C(n,0) = 1.
From the second equation, we get:
\(C(n,1)(px)^1 = 20x\)
=> C(n,1) * p = 20
From the third equation, we get:
\(C(n,2)(px)^2 = 160x^2\)
\(= > C(n,2) * p^2 = 160\)
We can simplify the last equation to get:
\(C(n,2) = 80 / p^2\)
Using the formula for binomial coefficients, we can write:
C(n,2) = n! / (2! * (n-2)!)
Simplifying this expression, we get:
C(n,2) = n(n-1) / 2
Substituting this in the earlier equation, we get:
\(n(n-1) / 2 = 80 / p^2\)
Multiplying both sides by \(2/p^2\) and simplifying, we get:
\(n(n-1)p^2 = 160\)
From the second equation, we have C(n,1) * p = 20, which gives us p = 20 / C(n,1).
Since C(n,1) = n, we have:
p = 20/n
Substituting this in the last equation, we get:
\(n(n-1)(20/n)^2 = 160\)
Simplifying, we get:
n(n-1) = 4
The only integer values of n that satisfy this equation are n=2 and n=3. We can check that n=2 does not work, but n=3 does work.
If we substitute n=3 in the equation for p, we get:
p = 20/3
Therefore, the values of n and p are n=3 and p=20/3, respectively.
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2 and 1/6 divided by 4 and 1/3, help pls
Answers
Answer:
The answer is 1 9/19
Step-by-step explanation:
How are slope and derivatives related.
Answers
Answer: When you plug in an x-value into a function's derivative, the y-values you get back FROM THE DERIVATIVE tell you the slope of a tangent line to the original function at that value of x.
Step-by-step explanation:
If 6% of a number is 48, find 1% of that number.
Answers
Answer:
8
Step-by-step explanation:
0.06/48=800 0.01x800=8
Imagine you deposited $500 in a savings account that has an annual interest rate of 5% for 5 years. Use the simple interest formula from the article to calculate how much you would earn each year. Then, calculate how much money you would have at the end of each year.
Answers
Answer:
After investing for 5 years at 5% interest, your $500 investment will have grown to $638.14
Step-by-step explanation:
George walked on a treadmill 16 ¾ miles over 4 days. How many miles did he walk per day?
Answers
Answer:
64 3/16 miles
Step-by-step explanation:
Total distance walked = 16 3/4 miles = 67/4 miles (converted to improper fraction to help with division later)
number of days walked = 4 days
Miles walked per day = total distance walked ÷ number of days
= 16 3/4 ÷ 4
= 67/4 ÷ 4
= 67/4 x 1/4
= 67/16
= 64 3/16 miles
a Ferris wheel has a radius of 65 ft what is the circumference of the circle ( this would be the distance traveled by a passenger in one full rotation of the Ferris wheel use pi equals 3
never mind it equals 390 ft because C= 2 x 3 x 65 witch equals 390 ft
Answers
The circumference of the Ferris is 408.2 ft.
What is the circumference?We know that the circumference can be described as the distance round the circle. We have to note that the radius of the feriis would be very important as we try to obtain the circumference of the ferris.
We can see that the radius of the Ferris as is clear from the question that we have here is 65 ft. When the circumference can be gotten from the use of;
C = 2πr
Where r = radius and C = circumference
Thus;
C = 2 * 3.14 * 65
C = 408.2 ft
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Please answer this question now
Answers
Answer:
420 cubic inches is the answer
Answer:
420 cubic inches
Step-by-step explanation:
Volume of rectangular pyramid
= 1/3*lbh
= 1/3 * 10*9*14
= 10*3*14
= 420 cubic inches
a polyhedron has all faces triangles or quadrilaterals, and $1001$ edges. what is the difference between the maximum and minimum possible numbers of faces?
Answers
Using Euler'formula, the difference between maximum and minimum possible numbers of faces is 96.
A polyhedron is a 3D shape with flat faces, straight edges, and sharp vertices (corners). "polyhedron" meaning "many" and "polyhedron" meaning "area". Therefore, when many planes are joined, they form a polyhedron.
Because all faces are triangles. So on a triangular base with triangular faces on both sides that meet at the vertex.
Therefore, the minimum number of triangular faces of a regular polyhedron is = 4
Next, the double pyramid has "2n" triangular faces. where n is a natural number greater than 2 and 'n' represents the number of sides of the base of the pyramid.
A polyhedron having equal quadrilateral faces is known as regular hexahedron.
quadrilateral faced polyhedron with total sides 100 and vertices 6 , then using Euler's formula
F + V-E = 2
=> F + 6- 100= 2 => F = 96 maximum faces possible = 96
Now the difference between maximum and minimum number of faces = 100 - 4 = 96
So the difference between maximum and minimum faces is 96.
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please help me on this! Its due tomorrow.
Answers
b. 231 ounces of nuts
the height of a golf ball after it is hit can be modeled by the equation where represents the number of seconds after the ball is hit, and represents the golf ball height, in feet. a) what is the height of the golf ball at the instant it is hit? feet b) what is the height of the golf ball 2.8 seconds after it is hit? feet c) how long does it take the ball to hit the ground after it is hit? (round answer to three decimal places.) seconds submit question question 3
Answers
a. The height of the golf ball at the instant it is hit is 0 feet.
b. The height of the golf ball 2.8 seconds after it is hit is approximately 62.72 feet.
c. It takes 4 seconds for the golf ball to hit the ground after it is hit.
a) To find the height of the golf ball at the instant it is hit, we need to evaluate the function h when t = 0.
Substituting t = 0 into the given equation, we get:
h = -16(0)² + 64(0) = 0
Therefore, the height of the golf ball at the instant it is hit is 0 feet.
b) To find the height of the golf ball 2.8 seconds after it is hit, we need to evaluate the function h when t = 2.8.
Substituting t = 2.8 into the given equation, we get:
h = -16(2.8)² + 64(2.8) ≈ 62.72
Therefore, the height of the golf ball 2.8 seconds after it is hit is approximately 62.72 feet.
c) To find how long it takes the ball to hit the ground after it is hit, we need to find the value of t when the height is 0.
Setting h = 0 in the given equation, we get:
0 = -16t² + 64t
0 = t(-16t + 64)
t = 0 or t = 4
Since the golf ball was hit upwards, we are only interested in the positive value of t, which is t = 4.
Therefore, it takes 4 seconds for the golf ball to hit the ground after it is hit.
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The question is -
The height in feet of a golf ball hit into the air is given by h = − 16t² + 64t, where t is the number of seconds elapsed since the ball was hit, and represents the golf ball height, in feet.
a) what is the height of the golf ball at the instant it is hit?
b) what is the height of the golf ball 2.8 seconds after it is hit?
c) how long does it take the ball to hit the ground after it is hit?
mai has a jar of quarters and dimes. she takes at least 10 coins out of the jar and has less than $2.00. write a system of inequalities that represents the number of quarters, `x`, and the number of dimes, `y`, that mai could have.
Answers
The system of inequalities that represents the number of quarters, x, and the number of dimes, y, that Mai could have is given by:
x + y ≥ 10 and 0.25x + 0.1y < 2
These are the two systems of inequalities that represent the number of quarters, x, and the number of dimes, y, that Mai could have.
Let x be the number of quarters and y be the number of dimes that Mai has. Then, the system of inequalities can be represented as:
Thus, the first inequality is x + y ≥ 10.
Also, Mai has less than $2.00, therefore, the second inequality is 0.25x + 0.1y < 2. The value of x and y are assumed to be non-negative integers.+
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PLS HELP MEMEMEMEME
Determine if triangle RST and triangle UVW are or are not similar, and, if they are, state how you know. (Note that figures are NOT necessarily drawn to scale.)
Answers
The two triangles ΔRST and ΔUVW are similar to each other by the SAS property.
What is the similarity?Two objects are said to be comparable if they have the same shape. Therefore, two figures are said to be comparable in mathematics if they share the same shapes, lines, or angles.
Given that there are two triangles ΔRST and ΔUVW. ΔRST has the angle ∠T = 59° and the sides RT = 15 units and ST = 13 units. The other triangle ΔWVU has the angle ∠W = 59° and the sides WV= 52 units and WU= 60 units.
Firstly the two angles are the same,
∠T= ∠W = 59°
The two sides are dilated by a scale factor of 4,
WV / ST = 52 / 13 = 4
WU / RT = 60 / 15 = 4
So these sides are similar.
Hence, from the SAS property, the two triangles are similar.
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What is the value of the expression 3a-b/6(b+a) when = 5 and b = -3?
Show your work to support your answer.
Answers
Step-by-step explanation:
3(5)-(-3)/6{(-3)+(5)}
15+3/6×2
6 is in denominator it'll cancelled by 2
18/3= 6 Answer
hope its clear
Answer:
Answer is 6
Step-by-step explanation:
If you like my answer than please mark me brainliest thanks
a presentation aid which is a pictorial representation of statistical data is called a
Answers
A presentation aid which is a pictorial representation of statistical data is called a Graph. Graph is defined as a statistical diagram or chart which is used to represent statistical data in an easy-to-understand format.
Graphs are very effective in helping people understand large quantities of complex data.Graphs can be used to represent different kinds of data such as line graphs, bar graphs, pie charts, scatter graphs, and more. Line graphs are used to show how a variable changes over time, bar graphs are used to compare different quantities, pie charts are used to show the proportion of different parts of a whole, and scatter graphs are used to show how two variables are related to each other.Graphs have a number of benefits over tables when it comes to representing data.
For one thing, they are often easier to read and understand than tables. They are also more visually appealing, which makes them more likely to grab people's attention. Finally, they can be used to show trends and relationships in data that would be difficult to see in a table.
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If the coefficient of determination is 0.81, the correlation coefficient (A) is 0.6561 (C) must be positive (B) could be either +0.9 or -0.9 (D) must be negative
Answers
For a R-squared of 0.81, the correlation coefficient (A) must be positive and can be either +0.9 or -0.9.
The coefficient of determination (R-squared) measures the proportion of variation in the dependent variable that is explained by the independent variables. It ranges from 0 to 1, with 0 indicating no linear relationship and 1 indicating a perfect linear relationship.
The coefficient of determination is 0.81, meaning that approximately 81% of the variation in the dependent variable can be explained by the independent variables. The correlation coefficient (A) is the square root of the coefficient of determination, A = \(\sqrt{0.81}\)= 0.9.
However, it is important to note that correlation coefficients are either positive or negative, indicating the direction of the relationship between variables. In this case, the coefficient of determination is positive, so the correlation coefficient (A) must also be positive. So the correct answer is (B). The correlation coefficient can be either +0.9 or -0.9, but it should be positive because the coefficient of determination is positive. Choice (D) that the correlation coefficient must be negative is incorrect in this context.
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PLEASE HELP IF WRONG WILL REPORT!
What is the measure of the unknown angle?
A. 145°
B.148°
C. 150°
D. 151°
Answers
Answer:
D
Step-by-step explanation:
Because the angle with measure 29 degrees and n are a linear pair, their value must add up to 180 degrees. Therefore:
n+29=180
n=180-29
n=151 degrees, or answer choice D
Hope this helps!
A town has a population of 2000 and grows at 4% every year.
What would the exponential function be if you were trying to predict what the
town's population would be after 12 years?
A 2000(04)12
B 2000(0.4)12
C 2000(1.04)12 D 0.04(2000)12
Answers
Answer:
\( \frac{4}{?100} \times \frac{2000}{1?} = 80 \\ \\ 2000 + 80 = 2080 \\ \\ \\ 2080 \times 12 = 24960\)
If x = -3, then x 2-7x + 10 equals
Answers
Answer: 40
Step-by-step explanation: x=−3
(−3)2−(7)(−3)+10
(−3)2−(7)(−3)+10
=40
A minimum element is deleted from a (min) binary heap with N elements. The running time worst case of this operation is
a. O(N)
b. O(N2)
c. O(logN)
d. O(NlogN)
Answers
The running time wοrst case οf deleting a minimum element frοm a (min) binary heap with N elements is O(lοgN). Therefοre, the cοrrect answer is c. O(lοgN).
What happens when deleting minimum element?When deleting the minimum element frοm a binary heap, the heap needs tο be restructured tο maintain its heap prοperty. This restructuring prοcess invοlves mοving elements within the heap and pοtentially swapping elements tο maintain the heap's structure and οrdering.
Since a binary heap is a cοmplete binary tree and has a height οf lοgN, the wοrst-case running time fοr deleting the minimum element is prοpοrtiοnal tο the height οf the heap, which is O(lοgN). This is because the number οf cοmparisοns and swaps required during the restructuring prοcess is dependent οn the height οf the heap.
Therefοre, the cοrrect answer is c. O(lοgN).
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Function graphing
Sketch a graph of the function f(x) = - 5 sin 6 5 4 3 2 -&t -7n -65-4n -3n-2n - j -2 -3 -4 -5 -6 + - (a) 27 3 4 5 \ / 67 8
Answers
To sketch the graph of the function `f(x) = - 5 sin 6 5 4 3 2 -&t -7n -65-4n -3n-2n - j -2 -3 -4 -5 -6 + - (a) 27 3 4 5 \ / 67 8`, we first need to identify its key features, which are:Amplitude = 5
Period = 2π/6
= π/3
Phase Shift = 2
The graph of the function `f(x) = - 5 sin 6x + 2` can be obtained by starting with the standard sine graph and making the following transformations:Reflecting it about the x-axis by multiplying the entire function by -1.
Multiplying the entire function by 5 to increase the amplitude.
Shifting the graph to the right by 2 units.For the specific domain provided in the question, we have:27 < 6x + 2 < 67 or 25/6 < x < 65/6.
This gives us a range of approximately 4.17 ≤ x ≤ 10.83.
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So, your task this week is a maths problem;
The distance between Exeter and London is 175 miles. Sam left Exeter at 10:00 on Tuesday for London. Morgan left London for Exeter at 13:00 the same day. They travelled on the same road. Up to the time when they met, Sam's average speed was 25 miles per hour, and Morgan's average speed was 35 miles an hour.
At what time did Sam and Morgan meet?
Answers
Answer: Sam and Morgan met up at 18:00. Sam left Exeter at 10:00, and was on the road for 7 hours. I know this because 175 / 25 = 7. 10:00 + 7:00 = 17:00 meaning Sam got there at 17:00. Morgan on the other hand left at 13:00 and had an average speed of 35 mph. 175/35=5. 13:00 + 5:00 = 18:00. By 18:00 both Sam and Morgan had arrived in London thus 18:00 us your answer.
Step-by-step explanation:
In each part, find exact values for the first four partial sums, find a closed form for the nth partial sum, and determine whether the series converges by calculating the limit of the nth partial sum. If the series converges, then state its sum. 3 - 14 Determine whether the series converges, and if so find its sum.
Answers
The limit of the nth partial sum does not exist, as it does not approach a finite value. Since the limit of the nth partial sum does not exist, the series does not converge.
To determine whether the series converges and find its sum, let's examine the given series:
3 - 14 + ...
The series alternates between the terms 3 and -14. We can calculate the partial sums to see if they approach a finite value as we add more terms.
First, let's find the first four partial sums:
Partial Sum 1: 3
Partial Sum 2: 3 - 14 = -11
Partial Sum 3: 3 - 14 + 3 = -8
Partial Sum 4: 3 - 14 + 3 - 14 = -22
Now, let's try to find a closed form for the nth partial sum. From the pattern observed, we can see that the sign alternates between positive and negative, and each term is either 3 or -14.
If we consider the index of the term, we can notice that when the index is odd, the term is positive (3), and when the index is even, the term is negative (-14). We can express this pattern using the following formula:
nth partial sum = (3 for odd n) - (14 for even n)
Using this formula, we can determine that the nth partial sum is (-14)^(n/2) * (-1)^(n/2) * (7/2) + 3/2.
Now, let's calculate the limit of the nth partial sum as n approaches infinity to determine if the series converges:
lim(n→∞) (-14)^(n/2) * (-1)^(n/2) * (7/2) + 3/2
As n increases, the term (-14)^(n/2) oscillates between positive and negative values, and (-1)^(n/2) also alternates between 1 and -1. Therefore, the limit of the nth partial sum does not exist, as it does not approach a finite value.
Since the limit of the nth partial sum does not exist, the series does not converge.
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Mookie betts of the boston red sox had the highest batting average for the 2018 major league baseball season. His average was 0.364. So, the likelihood of his getting a hit is 0.364 for each time he bats. Assume he has seven times at bat tonight in the red sox-yankee game. This is an example of what type of probability?
Answers
An example where Mookie betts of the boston red sox had the highest batting average in 2018 with probability of average batts is 0.364 for each is a binomial Probability example.
In the statistics, the concept of probability is handy in figuring out the likelihood a particular outcome would occur for an event or decision. On the basis of probability, one can figure out other related probabilities like the probability of not having that particular outcome or the probability of having only that outcome every time. There is Mookie betts of the boston red sox had the highest batting average in league baseball season.
The average probability = 0.364 for each time of bats.
That is Probability of hitting or success, p = 0.364
Number of trials, n = 7
That is Probability distribution is written as \(X \: \tilde \: \: Binom( n,p) \).
After reading all seniror, the probability for seven seven times at bat tonight in the red sox-yankee game is an example of binomial Probability.
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Please can someone help me !
Answers
Answer:
A
Step-by-step explanation:
pretty sure 99 percent sure
Answer:
It is the last one because they share similar sides based on how the triangle was created
It is generally suggested that the sample size in developing a multiple regression model should be at least four times the number of independent variables. Seleccione una: O Verdadero O Falso
Answers
False. It is not generally suggested that the sample size in developing a multiple regression model should be at least four times the number of independent variables.
There is no specific rule or guideline that states the sample size in developing a multiple regression model should be at least four times the number of independent variables. The appropriate sample size for a multiple regression model depends on various factors, such as the desired level of statistical power, the effect size, and the level of significance. In general, a larger sample size is preferred as it can increase the statistical power and reliability of the results.
However, the relationship between sample size and the number of independent variables is not fixed at a specific ratio like four times. It is important to consider the specific context of the study and the research question when determining the appropriate sample size for a multiple regression model.
Therefore, it is not accurate to suggest that the sample size should be at least four times the number of independent variables.
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