A univariate Gaussian or normal distributions can be completely determined by its mean and variance. Gaussian distributions can be applied to a large numbers of problems because of the central limit theorem (CLT). The CLT posits that when a large number of independent and identically distributed (i.i.d.) random variables are added, the cumulative distribution function (cdf) of their sum is approximated by the cdf of a normal distribution. Recall the probability density function of the univariate Gaussian with mean u and variance o?, N (1, 02): fx (2) = 1 27102 (2–)*/(20) e Probability review: PDF of Gaussian distribution 2 points possible (graded) In practice, it is not often that you will need to work directly with the probability density function (pdf) of Gaussian variables. Nonetheless, we will make sure we know how to manipulate the (pdf) in the next two problems. The pdf of a Gaussian random variable X is given by n n2(x – 2) fx (2) exp 327 then what is the mean p and variance o2 of X? (Enter your answer in terms of n.) M= 02 Probability review: PDF of Gaussian distribution 1 point possible (graded) Let X~ N (4,02), i.e. the pdf of X is 1 (x – u)? fx (x) exp 027 202 Let Y = 2X. Write down the pdf of the random variable Y. (Your answer should be in terms of y, o and p. Type mu for sigma for o.) fy (y) = Argmax 1 point possible (graded) Let fx (x;j, o?) denote the probability density function of a normally distributed variable X with mean u and variance 02. What value of x maximizes this function? (Enter mu for the mean ji, and sigma-2 for the variance o?.)

A univariate Gaussian or normal distributions can be completely determined by its mean and variance. Gaussian distributions can be applied to a large numbers of problems because of the central…

Continue Reading A univariate Gaussian or normal distributions can be completely determined by its mean and variance. Gaussian distributions can be applied to a large numbers of problems because of the central limit theorem (CLT). The CLT posits that when a large number of independent and identically distributed (i.i.d.) random variables are added, the cumulative distribution function (cdf) of their sum is approximated by the cdf of a normal distribution. Recall the probability density function of the univariate Gaussian with mean u and variance o?, N (1, 02): fx (2) = 1 27102 (2–)*/(20) e Probability review: PDF of Gaussian distribution 2 points possible (graded) In practice, it is not often that you will need to work directly with the probability density function (pdf) of Gaussian variables. Nonetheless, we will make sure we know how to manipulate the (pdf) in the next two problems. The pdf of a Gaussian random variable X is given by n n2(x – 2) fx (2) exp 327 then what is the mean p and variance o2 of X? (Enter your answer in terms of n.) M= 02 Probability review: PDF of Gaussian distribution 1 point possible (graded) Let X~ N (4,02), i.e. the pdf of X is 1 (x – u)? fx (x) exp 027 202 Let Y = 2X. Write down the pdf of the random variable Y. (Your answer should be in terms of y, o and p. Type mu for sigma for o.) fy (y) = Argmax 1 point possible (graded) Let fx (x;j, o?) denote the probability density function of a normally distributed variable X with mean u and variance 02. What value of x maximizes this function? (Enter mu for the mean ji, and sigma-2 for the variance o?.)

A unit vector is a vector with length, or norm, 1. Given any vector x, what is the unit vector pointing in the same direction as x? (Enter norm(x) for the norm ∥x∥ of the vector x. In general, you can enter the norm of a vector using the norm function).

A unit vector is a vector with length, or norm, 1. Given any vector x, what is the unit vector pointing in the same direction as x? (Enter norm(x) for…

Continue Reading A unit vector is a vector with length, or norm, 1. Given any vector x, what is the unit vector pointing in the same direction as x? (Enter norm(x) for the norm ∥x∥ of the vector x. In general, you can enter the norm of a vector using the norm function).

A hyperplane in n dimensions is a n – 1 dimensional subspace. For instance, a hyperplane in 2-dimensional space can be any line in that space and a hyperplane in 3-dimensional space can be any plane in that space. A hyperplane separates a space into two sides. In general, a hyperplane in n-dimensional space can be written as 0 01x+0zx2 + ‘.. + 0nx, = 0. For example, a hyperplane in two dimensions, which is a line, can be expressed as Ax Bx2 + C = 0. Using this representation of a plane, we can define a plane given an n-dimensional vector 0= and offset 00. This vector and offset combination would define the plane 0o +01×1 + 02×2 + -.. + 0nx, = 0. One feature of this representation is that the vector 0 is normal to the plane. 2. (a) 1 point possible (graded) Given a d-dimensional vector 0 and offset 0o which describe a hyperplane p, how many alternative descriptions e’ and e are there for p? 0 1 To check if a vector x is orthogonal to a plane p characterized by 0 and 00, we check whether x = ae for some a E R Ox0 0 x0 = 0

A hyperplane in n dimensions is a n - 1 dimensional subspace. For instance, a hyperplane in 2-dimensional space can be any line in that space and a hyperplane in…

Continue Reading A hyperplane in n dimensions is a n – 1 dimensional subspace. For instance, a hyperplane in 2-dimensional space can be any line in that space and a hyperplane in 3-dimensional space can be any plane in that space. A hyperplane separates a space into two sides. In general, a hyperplane in n-dimensional space can be written as 0 01x+0zx2 + ‘.. + 0nx, = 0. For example, a hyperplane in two dimensions, which is a line, can be expressed as Ax Bx2 + C = 0. Using this representation of a plane, we can define a plane given an n-dimensional vector 0= and offset 00. This vector and offset combination would define the plane 0o +01×1 + 02×2 + -.. + 0nx, = 0. One feature of this representation is that the vector 0 is normal to the plane. 2. (a) 1 point possible (graded) Given a d-dimensional vector 0 and offset 0o which describe a hyperplane p, how many alternative descriptions e’ and e are there for p? 0 1 To check if a vector x is orthogonal to a plane p characterized by 0 and 00, we check whether x = ae for some a E R Ox0 0 x0 = 0

1. (a) 3 points possible (graded) Note on notation: In this course, we will use regular letters as symbols for numbers, vectors, matrices, planes, hyperplanes, etc. You will need to distinguish what a letter represents from the context. Recall the dot product of a pair of vectors a and b -El-Fi Go aba2b:*.+ aqbn a b where a= and b= When thinking about a and bas vectors in n-dimensional space, we can also express the dot product as a.b= lall1에 cos a, where of a: is the angle formed between the vectors a and bin n-dimensional Euclidean space. Here, a refers to the length, also known norm. a| 0.4 What is the length of the vector 0.3 -0.15 What is the length of the vector 0.2 0.4 What is the angle (in radians) between a -0.15 ? Choose the answer that Ilies between 0 and . and 0.2 1. (b) 1 point possible (graded) a Given 3-dimensional vectors r – when is z) orthogonal to r2), 1.e. the angle between them is T/2? and z when 2a +2a3 when a- a + a = 0 when afa a = 0 STANDARD NOTATION Submit You have used 0 of 2 attempts 1. (c) 1 point possible (graded) , Given any vector z, what unit vector is a vector with length, or norm. the unit vector pointing in the same direction (Enter norm(x) for the norm l|z| of the vector z. In general, you can enter the norm of a vector using the norm function).

1. (a) 3 points possible (graded) Note on notation: In this course, we will use regular letters as symbols for numbers, vectors, matrices, planes, hyperplanes, etc. You will need to…

Continue Reading 1. (a) 3 points possible (graded) Note on notation: In this course, we will use regular letters as symbols for numbers, vectors, matrices, planes, hyperplanes, etc. You will need to distinguish what a letter represents from the context. Recall the dot product of a pair of vectors a and b -El-Fi Go aba2b:*.+ aqbn a b where a= and b= When thinking about a and bas vectors in n-dimensional space, we can also express the dot product as a.b= lall1에 cos a, where of a: is the angle formed between the vectors a and bin n-dimensional Euclidean space. Here, a refers to the length, also known norm. a| 0.4 What is the length of the vector 0.3 -0.15 What is the length of the vector 0.2 0.4 What is the angle (in radians) between a -0.15 ? Choose the answer that Ilies between 0 and . and 0.2 1. (b) 1 point possible (graded) a Given 3-dimensional vectors r – when is z) orthogonal to r2), 1.e. the angle between them is T/2? and z when 2a +2a3 when a- a + a = 0 when afa a = 0 STANDARD NOTATION Submit You have used 0 of 2 attempts 1. (c) 1 point possible (graded) , Given any vector z, what unit vector is a vector with length, or norm. the unit vector pointing in the same direction (Enter norm(x) for the norm l|z| of the vector z. In general, you can enter the norm of a vector using the norm function).

Asymptotics and Trends 0.0/4.0 points (graded) For each of the following functions f (2) below: • Find its limits lim f(c) as x approachs too. 1-too • Choose the values of x where f(c) is differentiable, i.e. f’ (2) exists • Choose the values of x where f (x) is also strictly increasing, i.e. f’ (2) > 0. 1. For f(x) = max(0, 2): (If the limit diverges to infty, enter inf for oo, and -inf for – ) lim f(x) = lim f() = 1+0o Choose the intervals of x where f(x) differentiable: f'(x) > 0: (Choose all that apply.) 2 0 (Graph this function on a piece of paper!) 2. For f(x) = 1 1+e-2 (Enter inf for and similarly -inf for – if the limit diverges to infty.) lim f(x) = 1-00 lim f(x) = Too Choose the intervals of x where f'(x) > 0 f (2) differentiable: (Choose all that apply.) 2 0 (Graph this function on a piece of paper!)

Asymptotics and Trends 0.0/4.0 points (graded) For each of the following functions f (2) below: • Find its limits lim f(c) as x approachs too. 1-too • Choose the values…

Continue Reading Asymptotics and Trends 0.0/4.0 points (graded) For each of the following functions f (2) below: • Find its limits lim f(c) as x approachs too. 1-too • Choose the values of x where f(c) is differentiable, i.e. f’ (2) exists • Choose the values of x where f (x) is also strictly increasing, i.e. f’ (2) > 0. 1. For f(x) = max(0, 2): (If the limit diverges to infty, enter inf for oo, and -inf for – ) lim f(x) = lim f() = 1+0o Choose the intervals of x where f(x) differentiable: f'(x) > 0: (Choose all that apply.) 2 0 (Graph this function on a piece of paper!) 2. For f(x) = 1 1+e-2 (Enter inf for and similarly -inf for – if the limit diverges to infty.) lim f(x) = 1-00 lim f(x) = Too Choose the intervals of x where f'(x) > 0 f (2) differentiable: (Choose all that apply.) 2 0 (Graph this function on a piece of paper!)

Write the derivative with respect to t for each of the following expressions: (a) In (1) (b) exp (2a/out) (, , and are constants) (c) sin (or?) (a is a constant) (d) logo (0/1) (a is a constant) (e) Int+ (1 – 1) In (1 – ) (6) exp(-E/R) (R and all E are constants) (8) “sin (+ – 2)dz (o and z are constants)

Write the derivative with respect to t for each of the following expressions: (a) In (1) (b) exp (2a/out) (, , and are constants) (c) sin (or?) (a is a…

Continue Reading Write the derivative with respect to t for each of the following expressions: (a) In (1) (b) exp (2a/out) (, , and are constants) (c) sin (or?) (a is a constant) (d) logo (0/1) (a is a constant) (e) Int+ (1 – 1) In (1 – ) (6) exp(-E/R) (R and all E are constants) (8) “sin (+ – 2)dz (o and z are constants)

Determine which of the following expressions are exact differentials: (a) df -(y + 2)dx + (2 + y)dy + (x + y)dz (b) df = 2xy dx + 2yzº dy – (1 – x? – 2yºz)dz (c) df – z dx + x dy + y dz (d) df -(x + 3x”yz)dx + (y + 2x”yz)dy + (x’y?)dz

Determine which of the following expressions are exact differentials: (a) df -(y + 2)dx + (2 + y)dy + (x + y)dz (b) df = 2xy dx + 2yzº dy…

Continue Reading Determine which of the following expressions are exact differentials: (a) df -(y + 2)dx + (2 + y)dy + (x + y)dz (b) df = 2xy dx + 2yzº dy – (1 – x? – 2yºz)dz (c) df – z dx + x dy + y dz (d) df -(x + 3x”yz)dx + (y + 2x”yz)dy + (x’y?)dz

Write down the total differential, df, for each of the following expressions: (a) f(x, y, z) = x2 + y2 + z2 (b) f(x, y, z) = sin (xyz) (c) f(x, y, z) = x3y – xy2

Write down the total differential, df, for each of the following expressions: (a) f(x, y, z) = x2 + y2 + z2 (b) f(x, y, z) = sin (xyz) (c)…

Continue Reading Write down the total differential, df, for each of the following expressions: (a) f(x, y, z) = x2 + y2 + z2 (b) f(x, y, z) = sin (xyz) (c) f(x, y, z) = x3y – xy2

The outside wall of a building 6 m high receives an average radiant heat flux from the sun of 1100 W/m2 . Assuming that 95 W/m2 is conducted through the wall, estimate the outside wall temperature. Assume the atmospheric air on the outside of the building is at 20°C.

The outside wall of a building 6 m high receives an average radiant heat flux from the sun of 1100 W/m2 . Assuming that 95 W/m2 is conducted through the…

Continue Reading The outside wall of a building 6 m high receives an average radiant heat flux from the sun of 1100 W/m2 . Assuming that 95 W/m2 is conducted through the wall, estimate the outside wall temperature. Assume the atmospheric air on the outside of the building is at 20°C.