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f X Y ( t , u ) = 3 88 ( 2 t + 3 u 2 ) for 0 t 2 , 0 u 1 + t (see Exercise 15 from "Problems on Random Vectors and Joint Distributions").

Z = I [ 0 , 1 ] ( X ) 4 X + I ( 1 , 2 ] ( X ) ( X + Y )

Determine P ( Z 2 )

P ( Z 2 ) = P ( Z Q = Q 1 M 1 Q 2 M 2 ) , where M 1 = { ( t , u ) : 0 t 1 , 0 u 1 + t }
M 2 = { ( t , u ) : 1 < t 2 , 0 u 1 + t }
Q 1 = { ( t , u ) : 0 t 1 / 2 } , Q 2 = { ( t , u ) : u 2 - t } (see figure)
P = 3 88 0 1 / 2 0 1 + t ( 2 t + 3 u 2 ) d u d t + 3 88 1 2 0 2 - t ( 2 t + 3 u 2 ) d u d t = 563 5632
tuappr Enter matrix [a b]of X-range endpoints [0 2] Enter matrix [c d]of Y-range endpoints [0 3] Enter number of X approximation points 200Enter number of Y approximation points 300 Enter expression for joint density (3/88)*(2*t + 3*u.^2).*(u<=1+t) Use array operations on X, Y, PX, PY, t, u, and PG = 4*t.*(t<=1) + (t+u).*(t>1); [Z,PZ]= csort(G,P); PZ2 = (Z<=2)*PZ' PZ2 = 0.1010 % Theoretical = 563/5632 = 0.1000
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Figure 1 is comprised of two separate figures, one for exercise 11, on the left, and one for exercise 12, on the right. The figure for exercise 11 is a cartesian graph of the first quadrant, with the horizontal axis labeled t, and the vertical axis labeled u. The values on the horizontal axis range from 0 to 2, in increments of 1, with an extra marking in between 0 and 1 for the value 0.5. The values on the vertical axis range from 0 to 3 in increments of 1. The figure contains three major shapes, with two smaller shaded shapes located inside a larger polygon. The large polygon has four sides. The first is a vertical portion located on the vertical axis, from (0, 0) to (0, 1). The upper side is a diagonal line, labeled u = 1 + t, that is drawn from (0, 1) to (2, 3). The rightmost side is another vertical line from the end of the second side at (2, 3) to the horizontal axis, at (2, 0). The bottom side of the shape connects (2, 0) horizontally to the origin, completing the shape. Inside this large polygon are two shaded shapes. The first is a right triangle with right-angle vertex located at (1, 0). One side is drawn vertically from (1, 0) to (1, 1), another is drawn horizontally from (1, 0) to (2, 0), and the hypotenuse is connected from (1, 1) to (2, 0). The hypotenuse is labeled u = 2 - t. There is a dashed vertical line connecting the triangles vertex at (1, 1) to the midpoint of the large polygon's diagonal side, at (1, 2). The second small shape is similar to the large polygon. It shares the same side on the vertical axis from (0, 0) to (0,1), and shares a portion of the diagonal line u = 1 + t from (0, 1) to (0.5, 1.5). Its third side is drawn vertically from that point to (0.5, 0) on the horizontal axis. A fourth side completes the polygon from (0.5, 0) to the origin.  The figure for exercise 12 is a cartesian graph of the first quadrant, with the horizontal axis labeled t, and the vertical axis labeled u. The values on the horizontal axis range from 0 to 2, in increments of 1, with an extra marking for the value 0.5. The vertical axis does not contain any markings for values. There are five lines, which, connecting to the horizontal and vertical axes, create four separate enclosed sections. Two of these sections are shaded. The outermost lines are labeled with equations. A horizontal line from (0, 1) to (1, 1) is labeled u = 1. A diagonal line with negative slope connects to the horizontal line and the t-axis from (1, 1) to (2, 0) and is labeled u = 2 - t. These two labeled lines, along with the u and t-axes, create a four-sided polygon with two horizontal sides, one vertical side on the u-axis, and one diagonal side of negative slope labeled u = 2 - t. There are three interior lines. The longest interior line extends from the origin to point (1, 1), and in doing so, divides the larger aforementioned polygon into two triangles. The first smaller interior line is drawn from (0.5, 1) to (0.5, 0.5), a vertical line. The second is drawn from (0.5, 0.5) to approximately (1.5, 0.5) a horizontal line. These two smaller interior lines divide the two aforementioned triangles into two smaller triangles, a large symmetric trapezoid, and a four-sided polygon. The polygon inside and the trapezoid are shaded, and the smaller triangles remain white. Figure 1 is comprised of two separate figures, one for exercise 11, on the left, and one for exercise 12, on the right. The figure for exercise 11 is a cartesian graph of the first quadrant, with the horizontal axis labeled t, and the vertical axis labeled u. The values on the horizontal axis range from 0 to 2, in increments of 1, with an extra marking in between 0 and 1 for the value 0.5. The values on the vertical axis range from 0 to 3 in increments of 1. The figure contains three major shapes, with two smaller shaded shapes located inside a larger polygon. The large polygon has four sides. The first is a vertical portion located on the vertical axis, from (0, 0) to (0, 1). The upper side is a diagonal line, labeled u = 1 + t, that is drawn from (0, 1) to (2, 3). The rightmost side is another vertical line from the end of the second side at (2, 3) to the horizontal axis, at (2, 0). The bottom side of the shape connects (2, 0) horizontally to the origin, completing the shape. Inside this large polygon are two shaded shapes. The first is a right triangle with right-angle vertex located at (1, 0). One side is drawn vertically from (1, 0) to (1, 1), another is drawn horizontally from (1, 0) to (2, 0), and the hypotenuse is connected from (1, 1) to (2, 0). The hypotenuse is labeled u = 2 - t. There is a dashed vertical line connecting the triangles vertex at (1, 1) to the midpoint of the large polygon's diagonal side, at (1, 2). The second small shape is similar to the large polygon. It shares the same side on the vertical axis from (0, 0) to (0,1), and shares a portion of the diagonal line u = 1 + t from (0, 1) to (0.5, 1.5). Its third side is drawn vertically from that point to (0.5, 0) on the horizontal axis. A fourth side completes the polygon from (0.5, 0) to the origin.  The figure for exercise 12 is a cartesian graph of the first quadrant, with the horizontal axis labeled t, and the vertical axis labeled u. The values on the horizontal axis range from 0 to 2, in increments of 1, with an extra marking for the value 0.5. The vertical axis does not contain any markings for values. There are five lines, which, connecting to the horizontal and vertical axes, create four separate enclosed sections. Two of these sections are shaded. The outermost lines are labeled with equations. A horizontal line from (0, 1) to (1, 1) is labeled u = 1. A diagonal line with negative slope connects to the horizontal line and the t-axis from (1, 1) to (2, 0) and is labeled u = 2 - t. These two labeled lines, along with the u and t-axes, create a four-sided polygon with two horizontal sides, one vertical side on the u-axis, and one diagonal side of negative slope labeled u = 2 - t. There are three interior lines. The longest interior line extends from the origin to point (1, 1), and in doing so, divides the larger aforementioned polygon into two triangles. The first smaller interior line is drawn from (0.5, 1) to (0.5, 0.5), a vertical line. The second is drawn from (0.5, 0.5) to approximately (1.5, 0.5) a horizontal line. These two smaller interior lines divide the two aforementioned triangles into two smaller triangles, a large symmetric trapezoid, and a four-sided polygon. The polygon inside and the trapezoid are shaded, and the smaller triangles remain white.

f X Y ( t , u ) = 24 11 t u for 0 t 2 , 0 u min { 1 , 2 - t } (see Exercise 17 from "Problems on Random Vectors and Joint Distributions").

Z = I M ( X , Y ) 1 2 X + I M c ( X , Y ) Y 2 , M = { ( t , u ) : u > t }

Determine P ( Z 1 / 4 ) .

P ( Z 1 / 4 ) = P ( ( X , Y ) M 1 Q 1 M 2 Q 2 ) , M 1 = { ( t , u ) : 0 t u 1 }
M 2 = { ( t , u ) : 0 t 2 , 0 t min ( t , 2 - t ) }
Q 1 = { ( t , u ) : t 1 / 2 } Q 2 = { ( t , u ) : u 1 / 2 } (see figure)
P = 24 11 0 1 / 2 0 1 t u d u d t + 24 11 1 / 2 3 / 2 0 1 / 2 t u d u d t + 24 11 3 / 2 2 0 2 - t t u d u d t = 85 176
tuappr Enter matrix [a b]of X-range endpoints [0 2] Enter matrix [c d]of Y-range endpoints [0 1] Enter number of X approximation points 400Enter number of Y approximation points 200 Enter expression for joint density (24/11)*t.*u.*(u<=min(1,2-t)) Use array operations on X, Y, PX, PY, t, u, and PG = 0.5*t.*(u>t) + u.^2.*(u<t); [Z,PZ]= csort(G,P); pp = (Z<=1/4)*PZ' pp = 0.4844 % Theoretical = 85/176 = 0.4830
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f X Y ( t , u ) = 3 23 ( t + 2 u ) for 0 t 2 , 0 u max { 2 - t , t } (see Exercise 18 from "Problems on Random Vectors and Joint Distributions").

Z = I M ( X , Y ) ( X + Y ) + I M c ( X , Y ) 2 Y , M = { ( t , u ) : max ( t , u ) 1 }

Determine P ( Z 1 ) .

P ( Z 1 ) = P ( ( X , Y ) M 1 Q 1 M 2 Q 2 ) , M 1 = { ( t , u ) : 0 t 1 , 0 u 1 - t }
M 2 = { ( t , u ) : 1 t 2 , 0 u t }
Q 1 = { ( t , u ) : u 1 - t } Q 2 = { ( t , u ) : u 1 / 2 } (see figure)
P = 3 23 0 1 0 1 - t ( t + 2 u ) d u d t + 3 23 1 2 0 1 / 2 ( t + 2 u ) d u d t = 9 46
tuappr Enter matrix [a b]of X-range endpoints [0 2] Enter matrix [c d]of Y-range endpoints [0 2] Enter number of X approximation points 300Enter number of Y approximation points 300 Enter expression for joint density (3/23)*(t + 2*u).*(u<=max(2-t,t)) Use array operations on X, Y, PX, PY, t, u, and PM = max(t,u)<= 1; G = M.*(t + u) + (1 - M)*2.*u;p = total((G<=1).*P) p = 0.1960 % Theoretical = 9/46 = 0.1957
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Figure two is comprised of two cartesian graphs, both displaying only the first coordinate. Both graphs use t as the horizontal axis and u as the vertical axis. Both axes on the graphs range in value from 0 to 2 in increments of 1. The first graph, for exercise 13,  contains five labeled lines, and two unlabeled lines. The lines are labeled with their equations. From (0, 2) to (1, 1) is a line segment labeled u = 2 - t. From (1, 1) to (2, 2) is a line segment labeled u = t. From (0, 1) to (1, 0) is a line segment labeled u = 1 - t. From (1, 0.5) to (2, 0.5) is a line segment labeled u = 0.5. From (2, 0) to (2, 2) is a line labeled t = 2. From (0, 1) to (1, 1) is an unlabeled line segment. From (1, 0) to (1, 1) is the final unlabeled line segment. Two shapes created by the lines on this graph are shaded. The first is a right triangle bound by u = 1 - t and the horizontal and vertical axes. The second is a rectangle bound by u = 0.5, t = 2, the vertical unlabeled line segment from (1, 0) to (1, 1), and the horizontal axis. The second graph, for exercise 14, contains 5 labeled line segments, one unlabeled line segment, and one dashed, unlabeled line segment. From (0, 2) to (1, 2) is a line segment labeled u = 2. From (0, 2) to (1, 1) is a line segment labeled u = 2 - t. From (1, 2) to (2, 1) is a line segment labeled u = 3 - t. From (1, 1) to (2, 1) is a line segment labeled u = 1. From (2, 1) to (2, 0) is a line segment labeled t = 2. The unlabeled solid line segment is drawn from (1, 2) to (1, 1). The unlabeled dashed line segment is drawn from (0, 1) to (1, 1). A large region, bounded by u = 2 - t, u = 1, t = 2, the vertical axis, and the horizontal axis, is shaded grey. Figure two is comprised of two cartesian graphs, both displaying only the first coordinate. Both graphs use t as the horizontal axis and u as the vertical axis. Both axes on the graphs range in value from 0 to 2 in increments of 1. The first graph, for exercise 13,  contains five labeled lines, and two unlabeled lines. The lines are labeled with their equations. From (0, 2) to (1, 1) is a line segment labeled u = 2 - t. From (1, 1) to (2, 2) is a line segment labeled u = t. From (0, 1) to (1, 0) is a line segment labeled u = 1 - t. From (1, 0.5) to (2, 0.5) is a line segment labeled u = 0.5. From (2, 0) to (2, 2) is a line labeled t = 2. From (0, 1) to (1, 1) is an unlabeled line segment. From (1, 0) to (1, 1) is the final unlabeled line segment. Two shapes created by the lines on this graph are shaded. The first is a right triangle bound by u = 1 - t and the horizontal and vertical axes. The second is a rectangle bound by u = 0.5, t = 2, the vertical unlabeled line segment from (1, 0) to (1, 1), and the horizontal axis. The second graph, for exercise 14, contains 5 labeled line segments, one unlabeled line segment, and one dashed, unlabeled line segment. From (0, 2) to (1, 2) is a line segment labeled u = 2. From (0, 2) to (1, 1) is a line segment labeled u = 2 - t. From (1, 2) to (2, 1) is a line segment labeled u = 3 - t. From (1, 1) to (2, 1) is a line segment labeled u = 1. From (2, 1) to (2, 0) is a line segment labeled t = 2. The unlabeled solid line segment is drawn from (1, 2) to (1, 1). The unlabeled dashed line segment is drawn from (0, 1) to (1, 1). A large region, bounded by u = 2 - t, u = 1, t = 2, the vertical axis, and the horizontal axis, is shaded grey.

f X Y ( t , u ) = 12 179 ( 3 t 2 + u ) , for 0 t 2 , 0 u min { 2 , 3 - t } (see Exercise 19 from "Problems on Random Vectors and Joint Distributions").

Z = I M ( X , Y ) ( X + Y ) + I M c ( X , Y ) 2 Y 2 , M = { ( t , u ) : t 1 , u 1 }

Determine P ( Z 2 ) .

P ( Z 2 ) = P ( ( , Y ) M 1 Q 1 ( M 2 M 3 ) Q 2 ) , M 1 = { ( t , u ) : 0 t 1 , 1 u 2 }
M 2 = { ( t , u ) : 0 t 1 , 0 u 1 } M 3 = { ( t , u ) : 1 t 2 , 0 u 3 - t }
Q 1 = { ( t , u ) : u 1 - t } Q 2 = { ( t , u ) : u 1 / 2 } (see figure)
P = 12 179 0 1 0 2 - t ( 3 t 2 + u ) d u d t + 12 179 1 2 0 1 ( 3 t 2 + u ) d u d t = 119 179
tuappr Enter matrix [a b]of X-range endpoints [0 2] Enter matrix [c d]of Y-range endpoints [0 2] Enter number of X approximation points 300Enter number of Y approximation points 300 Enter expression for joint density (12/179)*(3*t.^2 + u).*(u<=min(2,3-t)) Use array operations on X, Y, PX, PY, t, u, and PM = (t<=1)&(u>=1); Z = M.*(t + u) + (1 - M)*2.*u.^2;G = M.*(t + u) + (1 - M)*2.*u.^2; p = total((G<=2).*P) p = 0.6662 % Theoretical = 119/179 = 0.6648
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f X Y ( t , u ) = 12 227 ( 3 t + 2 t u ) , for 0 t 2 , 0 u min { 1 + t , 2 } (see Exercise 20 from "Problems on Random Variables and Joint Distributions")

Z = I M ( X , Y ) X + I M c ( X , Y ) Y X , M = { ( t , u ) : u min ( 1 , 2 - t ) }

Detemine P ( Z 1 ) .

Figure three, for exercise 15, is comprised of one cartesian graph, with only the first quadrant displayed. The horizontal axis is labeled t, and the vertical axis is labeled u. The values on both axes range from 0 to 2 in increments of 1. There are 6 labeled line segments, one unlabeled line segment, and two resulting shaded shapes in this figure. The line segment from (0, 1) to (1, 2) is labeled u = 1 + t. The line segment from (0, 1) to (1, 1) is labeled u = 1. The line segment from (1, 2) to (2, 2) is labeled u = 2. The line segment from (1, 1) to (2, 2) is labeled u = t. The line segment from (1, 1) to (2, 0) is labeled u = 2 - t. The line segment from (2, 0) to (2, 2) is labeled t = 2. The unlabeled line segment is drawn from (1, 0) to (1, 1). A shaded square resulting from the drawn segments is bound by u = 1, the vertical axis, the horizontal axis, and the unlabeled line segment. A shaded triangle is bound by u = t, u = 2 - t, and t = 2. Figure three, for exercise 15, is comprised of one cartesian graph, with only the first quadrant displayed. The horizontal axis is labeled t, and the vertical axis is labeled u. The values on both axes range from 0 to 2 in increments of 1. There are 6 labeled line segments, one unlabeled line segment, and two resulting shaded shapes in this figure. The line segment from (0, 1) to (1, 2) is labeled u = 1 + t. The line segment from (0, 1) to (1, 1) is labeled u = 1. The line segment from (1, 2) to (2, 2) is labeled u = 2. The line segment from (1, 1) to (2, 2) is labeled u = t. The line segment from (1, 1) to (2, 0) is labeled u = 2 - t. The line segment from (2, 0) to (2, 2) is labeled t = 2. The unlabeled line segment is drawn from (1, 0) to (1, 1). A shaded square resulting from the drawn segments is bound by u = 1, the vertical axis, the horizontal axis, and the unlabeled line segment. A shaded triangle is bound by u = t, u = 2 - t, and t = 2.
P ( Z 1 ) = P ( ( X , Y ) M 1 Q 1 M 2 Q 2 ) , M 1 = M , M 2 = M c
Q 1 = { ( t , u ) : 0 t 1 } Q 2 = { ( t , u ) : u t } (see figure)
P = 12 227 0 1 0 1 ( 3 t + 2 t u ) d u d t + 12 227 1 2 2 - t t ( 3 t + 2 t u ) d u d t = 124 227
tuappr Enter matrix [a b]of X-range endpoints [0 2] Enter matrix [c d]of Y-range endpoints [0 2] Enter number of X approximation points 400Enter number of Y approximation points 400 Enter expression for joint density (12/227)*(3*t+2*t.*u).*(u<=min(1+t,2)) Use array operations on X, Y, PX, PY, t, u, and PQ = (u<=1).*(t<=1) + (t>1).*(u>=2-t).*(u<=t); P = total(Q.*P)P = 0.5478 % Theoretical = 124/227 = 0.5463
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The class { X , Y , Z } is independent.

          X = - 2 I A + I B + 3 I C . Minterm probabilities are (in the usual order)

0 . 255 0 . 025 0 . 375 0 . 045 0 . 108 0 . 012 0 . 162 0 . 018

          Y = I D + 3 I E + I F - 3 . The class { D , E , F } is independent with

P ( D ) = 0 . 32 P ( E ) = 0 . 56 P ( F ) = 0 . 40

          Z has distribution

Value -1.3 1.2 2.7 3.4 5.8
Probability 0.12 0.24 0.43 0.13 0.08

Determine P ( X 2 + 3 X Y 2 > 3 Z ) .

% file npr10_16.m Data for [link] cx = [-2 1 3 0];pmx = 0.001*[255 25 375 45 108 12 162 18];cy = [1 3 1 -3];pmy = minprob(0.01*[32 56 40]);Z = [-1.3 1.2 2.7 3.4 5.8];PZ = 0.01*[12 24 43 13 8];disp('Data are in cx, pmx, cy, pmy, Z, PZ') npr10_16 % Call for dataData are in cx, pmx, cy, pmy, Z, PZ [X,PX]= canonicf(cx,pmx); [Y,PY]= canonicf(cy,pmy); icalc3Enter row matrix of X-values X Enter row matrix of Y-values YEnter row matrix of Z-values Z Enter X probabilities PXEnter Y probabilities PY Enter Z probabilities PZUse array operations on matrices X, Y, Z, PX, PY, PZ, t, u, v, and PM = t.^2 + 3*t.*u.^2>3*v; PM = total(M.*P)PM = 0.3587
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The simple random variable X has distribution

X = [ - 3 . 1 - 0 . 5 1 . 2 2 . 4 3 . 7 4 . 9 ] P X = [ 0 . 15 0 . 22 0 . 33 0 . 12 0 . 11 0 . 07 ]
  1. Plot the distribution function F X and the quantile function Q X .
  2. Take a random sample of size n = 10 , 000 . Compare the relative frequency for each value with the probability that value is taken on.
X = [-3.1 -0.5 1.2 2.4 3.7 4.9];PX = 0.01*[15 22 33 12 11 7];ddbn Enter row matrix of VALUES XEnter row matrix of PROBABILITIES PX % Plot not reproduced here dquanplotEnter VALUES for X X Enter PROBABILITIES for X PX % Plot not reproduced hererand('seed',0) % Reset random number generator dsample % for comparison purposesEnter row matrix of VALUES X Enter row matrix of PROBABILITIES PXSample size n 10000 Value Prob Rel freq-3.1000 0.1500 0.1490 -0.5000 0.2200 0.21641.2000 0.3300 0.3340 2.4000 0.1200 0.11843.7000 0.1100 0.1070 4.9000 0.0700 0.0752Sample average ex = 0.8792 Population mean E[X]= 0.859 Sample variance vx = 5.146Population variance Var[X] = 5.112
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Questions & Answers

what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive
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Source:  OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
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