AskSia

Plus

11. (a) Find a QR factorization of $A=\left(\begin{array}{lll}1 & 0 & 0 \\ 1 & 1...

Jan 7, 2024

11. (a) Find a QR factorization of

A=\left(\begin{array}{lll}1 &amp; 0 &amp; 0 \\ 1 &amp; 1 &amp; 0 \\ 1 &amp; 1 &amp; 1 \\ 1 &amp; 1 &amp; 1\end{array}\right)

Or (b) Find the least squares solution of the linear system

\mathrm{Ax}=\mathrm{b}

given by

x_{1}-x_{2}=4 ; 3 x_{1}+2 x_{2}=1 ;-2 x_{1}+4 x_{2}=3

12. (a) (i) For a certain binary, communication channel, the probability that a transmitted ' 0 ' is received as a ' 0 ' is 0.95 and the probability that a transmitted ' 1 ' is received as ' 1 ' is 0.90 . If the probability that a '

\alpha

' is transmitted is 0.4 , find the probability that (1) a ' 1 ' is received and (2) a '1' was transmitted given that a '1' was received. (ii) Find the moment generating function of Geometric distribution. Or (b) (i) Messages arrive at a switchboard in a Poisson manner at an average rate of six per hour. Find the probability for each of the following events: (1) exactly two messages arrive within one hour (2) no message arrives within one hour (3) at least three messages arrive within one hour. (ii) If

X=N(3,9)

, which means that

X

is Normal random variable with mean 3 and variance 9 , find the probability that

X

lies between 2 and 5. 13. (a) Find the bivariate probability distribution of

(X, Y)

given below, find

P(X \leq 1), P(Y \leq 3), P(X \leq 1, Y \leq 3), P(X \leq 1 / Y \leq 3)

and

P(X+Y \leq 4)

. \begin{tabular}{|c|c|c|c|c|c|c|} \hline

\mathrm{X}^{\mathrm{Y}}

& 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 0 & 0 & 0 &

1 / 32

2 / 32

2 / 32

3 / 32

\\ \hline 1 &

1 / 16

1 / 16

1 / 8

1 / 8

1 / 8

1 / 8

\\ \hline 2 &

1 / 32

1 / 32

1 / 64

1 / 64

& 0 &

2 / 64

\\ \hline \end{tabular} Or (b) Find the coefficient of correlation between

\mathrm{X}

and

\mathrm{Y}

, using following dats: \begin{tabular}{lllllllll}

\mathrm{X}

& 65 & 66 & 67 & 67 & 68 & 69 & 70 & 72 \\

\mathrm{Y}

& 67 & 68 & 65 & 68 & 72 & 72 & 69 & 71 \end{tabular}

Solution by Steps

step 1

To find the QR factorization of the matrix

A

, we first need to apply the Gram-Schmidt process to the columns of

A

step 2

Let

u_1, u_2, u_3

be the columns of

A

. We set

v_1 = u_1

step 3

Now, compute

v_2 = u_2 - \frac{\langle u_2, v_1 \rangle}{\langle v_1, v_1 \rangle}v_1

step 4

Compute

v_3 = u_3 - \frac{\langle u_3, v_1 \rangle}{\langle v_1, v_1 \rangle}v_1 - \frac{\langle u_3, v_2 \rangle}{\langle v_2, v_2 \rangle}v_2

step 5

Normalize

v_1, v_2, v_3

to get the orthonormal vectors

q_1, q_2, q_3

step 6

Form the matrix

Q

with columns

q_1, q_2, q_3

and the matrix

R

with the dot products of

u_i

and

q_j

as entries

Answer

The QR factorization of

A

A = QR

, where

Q

is an orthogonal matrix with columns

q_1, q_2, q_3

and

R

is an upper triangular matrix.

Key Concept

QR Factorization

Explanation

QR factorization involves decomposing a matrix into an orthogonal matrix

Q

and an upper triangular matrix

R

. This is achieved through the Gram-Schmidt process.

---

Solution by Steps

step 1

To find the least squares solution of the linear system

Ax = b

, we need to solve the normal equations

A^TA\hat{x} = A^Tb

step 2

Compute

A^TA

and

A^Tb

step 3

Solve the resulting system of equations for

\hat{x}

Answer

The least squares solution

\hat{x}

is the solution to the normal equations

A^TA\hat{x} = A^Tb

Key Concept

Least Squares Solution

Explanation

The least squares solution minimizes the sum of the squares of the residuals, the differences between the observed and predicted values. It is found by solving the normal equations.

---

Solution by Steps

step 1

To find the probability that a '1' is received, we need to consider both the probability that a '1' is transmitted and received correctly and the probability that a '0' is transmitted but received incorrectly as a '1'

step 2

Calculate the probability that a '1' is transmitted and received as a '1':

P(T=1) \times P(R=1|T=1)

step 3

Calculate the probability that a '0' is transmitted and received as a '1':

P(T=0) \times P(R=1|T=0)

step 4

Add the probabilities from step 2 and step 3 to get the total probability that a '1' is received

step 5

To find the probability that a '1' was transmitted given that a '1' was received, use Bayes' theorem:

P(T=1|R=1) = \frac{P(R=1|T=1) \times P(T=1)}{P(R=1)}

Answer

The probability that a '1' is received is the sum of the probabilities from step 2 and step 3. The probability that a '1' was transmitted given that a '1' was received is given by Bayes' theorem in step 5.

Key Concept

Probability and Bayes' Theorem

Explanation

To find the probability of an event given another event, we use Bayes' theorem, which relates the conditional and marginal probabilities of random events.

---

Solution by Steps

step 1

The moment generating function (MGF) of a Geometric distribution with success probability

p

is given by

M(t) = \frac{pe^t}{1-(1-p)e^t}

, for t < -\ln(1-p)

Answer

The MGF of a Geometric distribution is

M(t) = \frac{pe^t}{1-(1-p)e^t}

Key Concept

Moment Generating Function

Explanation

The MGF of a distribution is a function that encodes all the moments of the distribution and can be used to find them.

---

Solution by Steps

step 1

To find the probability of exactly two messages arriving within one hour, use the Poisson probability formula

P(X=k) = \frac{e^{-\lambda}\lambda^k}{k!}

where

\lambda

is the average rate and

k

is the number of occurrences

step 2

Calculate

P(X=2)

using

\lambda = 6

and

k = 2

step 3

To find the probability of no messages arriving within one hour, set

k = 0

and calculate

P(X=0)

step 4

To find the probability of at least three messages arriving within one hour, calculate

1 - P(X=0) - P(X=1) - P(X=2)

Answer

The probabilities are

P(X=2)

for exactly two messages,

P(X=0)

for no messages, and

1 - P(X=0) - P(X=1) - P(X=2)

for at least three messages.

Key Concept

Poisson Distribution

Explanation

The Poisson distribution is used to model the number of events occurring within a fixed interval of time or space, given the events occur with a known constant mean rate and independently of the time since the last event.

---

Solution by Steps

step 1

To find the probability that

X

lies between 2 and 5, we need to use the standard normal distribution

step 2

Convert

X

to the standard normal variable

Z

using

Z = \frac{X - \mu}{\sigma}

, where

\mu

is the mean and

\sigma

is the standard deviation

step 3

Find P(2 < X < 5) by calculating P\left(\frac{2 - 3}{3} < Z < \frac{5 - 3}{3}\right)

step 4

Use standard normal distribution tables or a calculator to find the probabilities P(Z < \frac{2}{3}) and P(Z < -\frac{1}{3}) , then subtract the latter from the former

Answer

The probability that

X

lies between 2 and 5 is P(Z < \frac{2}{3}) - P(Z < -\frac{1}{3}) .

Key Concept

Standard Normal Distribution

Explanation

To find the probability of a normal random variable lying between two values, we convert it to the standard normal form and use standard normal distribution tables or calculators.

---

Solution by Steps

step 1

To find the bivariate probability distribution of

(X, Y)

, we need to sum the probabilities in the given table for the respective events

step 2

Calculate

P(X \leq 1)

by summing the probabilities in the table where

X

is 0 or 1

step 3

Calculate

P(Y \leq 3)

by summing the probabilities in the table where

Y

is 1, 2, or 3

step 4

Calculate

P(X \leq 1, Y \leq 3)

by summing the probabilities in the table where

X

is 0 or 1 and

Y

is 1, 2, or 3

step 5

Calculate

P(X \leq 1 / Y \leq 3)

by dividing

P(X \leq 1, Y \leq 3)

P(Y \leq 3)

step 6

Calculate

P(X + Y \leq 4)

by summing the probabilities in the table where the sum of

X

and

Y

is 4 or less

Answer

The probabilities are

P(X \leq 1)

P(Y \leq 3)

P(X \leq 1, Y \leq 3)

P(X \leq 1 / Y \leq 3)

, and

P(X + Y \leq 4)

as calculated in the steps above.

Key Concept

Bivariate Probability Distribution

Explanation

A bivariate probability distribution gives the probability that a pair of random variables falls within a certain range. It is calculated by summing the probabilities of the events of interest.

---

Solution by Steps

step 1

To find the coefficient of correlation between

X

and

Y

, we need to calculate the means of

X

and

Y

\bar{X}

and

\bar{Y}

, respectively

step 2

Calculate the standard deviations of

X

and

Y

\sigma_X

and

\sigma_Y

, respectively

step 3

Calculate the covariance between

X

and

Y

\text{Cov}(X, Y)

step 4

The coefficient of correlation

r

is given by

r = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y}

Answer

The coefficient of correlation between

X

and

Y

r

, calculated as

r = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y}

Key Concept

Coefficient of Correlation

Explanation

The coefficient of correlation measures the strength and direction of the linear relationship between two variables. It is calculated using the means, standard deviations, and covariance of the variables.