Solve the following kindly

.

Suppose we throw a needle on a large sheet of paper, on which horizontal

lines are drawn, which are at needle-length apart (see also Exercise 21.16).

Choose one of the horizontal lines as x-axis, and let (X, Y ) be the center of the

needle. Furthermore, let Z be the distance of this center (X, Y ) to the nearest

horizontal line under (X, Y ), and let H be the angle between the needle and

the positive x-axis.

a. Assuming that the length of the needle is equal to 1, argue that Z has

a U(0, 1) distribution. Also argue that H has a U(0, ?) distribution and

that Z and H are independent.

b. Show that the needle hits a horizontal line when

Z ?

1

2 sin H or 1 ? Z ?

1

2 sin H.

c. Show that the probability that the needle will hit one of the horizontal

lines equals 2

(a) For the Bayesian model given by ( | )~ ( , ) Y Bin n ? ? and the prior

? ?? ~ (,) Beta , find the posterior predictive density of a future data

value x, whose distribution is defined by ( | , )~ ( , ) x y Bin m ? ? .

(b) A bent coin is tossed 20 times and 6 heads come up. Assuming a flat

prior on the probability of heads on a single toss, what is the probability

that exactly one head will come up on the next two tosses of the same

coin? Answer this using results in (a).Chapter 3: Bayesian Basics Part 3

123

(c) A bent coin is tossed 20 times and 6 heads come up. Assume a

Beta(20.3,20.3) prior on the probability of heads.

Find the expected number of times you will have to toss the same coin

again repeatedly until the next head comes up.

(d) A bent coin is tossed 20 times and 6 heads come up. Assume a

Beta(20.3,20.3) prior on the probability of heads.

Now consider tossing the coin repeatedly until the next head, writing

down the number of tosses, and then doing all of this again repeatedly,

again and again.

The result will be a sequence of natural numbers (for example

3, 1, 1, 4, 2, 2, 1, 5, 1, ....), where each number represents a number of

tails in a row within the sequence, plus one.

Next define ? to be the average of a very long sequence like this (e.g.

one of length 1,000,000). Find the posterior predictive density and mean

of ? (approximately)

In how many ways 5 boys and 3 girls can be seated in a row, so that no two girls are

together?

3. How many 6- digit telephone numbers can be constructed with the digits

0,1,2,3,4,5,6,7,8,9 if each numbers starts with 35 and no digit appear more than

once?

4. Find the number of arrangements that can be made out of the letters of the word

"ASSASSINATION"

5. (a) In how many ways can 8 identical beads be strung on a necklace?

(b) In how many ways can 8 boys form a ring?

6. Find the rank of the word 'CHAT' in dictionary.

LITY AND RANDOM VARIABLES Section1 Lecture (2 urses / PROBABILITY AND RANDOM VARIABLES Section1 Lecture (20194 110713120_AAUP - JENIN) / 25 September - 1 The distance that a player can through a ball to is defined by an exponential Random variable X with a=10 and b =10. FOX)=1-e-(x/10-1) u(x-10) Given that the distance is not more than 40, what is the probability that it is more than 15. Select one: O a. 0.5449 O b. 0.6563 O c. none of these O d. 0.3437 Next pa ct (4036) Jump to... Zoom finalprobability that this random variable will take on a value more than 3. Question 19 O out of 3 points Answer the following question for a random variable with values in the interval [0,5], whose density curve is shown below. 0.4 0 1 2 Find the probability that this random variable will take on a value less than or equal to 3. Question 20 0 out of 4 points 173 AM O e a 9 W 6.21/20202. Record the annual depreciation expense for the year 2018. Enter your computation here: 4. Assume that More Company sells the equipment on December 31, 2022 for $350,000. Record the entry to record the sale of the equipment. Use this space to show computation: Use this space to show computation: Use this space to show""""' ""J \"'J\"-"'J"'GJ""- "'"""'_'} Exercise 13. Suppose I tell you that the following list of 20 numbers is a random sample from a Gaussian random variable, but I don't tell you the mean or standard deviation. Also, around one or two of the numbers was corrupted by noise, computational error, tabulation error, etc., so that it is totally unrelated to the actual Gaussian random variable. 1.2045, 1.4329, O.3616, O.3743, 2.7298, 1.0601, 1.3298, 0.2554, 6.1365, 1.2185 2.7273, U.8453, 3.4282, 3.2270, 1.0137, 2.0553, 5.5393, 0.2572, 1.4512, 1.2347 To the best of your ability, determine what the mean and standard deviation are of this random variable. Supposing you had instead a billion numbers, and 5 or 10 percent of them were corrupted samples, can you come up with some automatic way of throwing out the corrupted samples? (Once again, there could be more than one right answer here; the question is intentionally open-ended.)