Statistics and Probability Answers

Show that

Ec[logƒ(Y)] = —— log(26²) —

7²+(m_µ)²

202

(c) Show that Kullback-Leibler divergence of f (y) with respect to g(y) is given by

D(glf) EG[logg (Y)]-EG[log f(Y)]

0² t² + (m-μ)² - 1}

Suppose that the true distribution g(y) generating data and the specified model f (y) have normal distribution N(m, t2) and N(u, o²), respectively.

(a) Show that

Ec[logg (Y)] = -log(2rt²) - 1

1

where EG[] is an expectation with respect to the true distribution N(m, t²).

If a bag contains errors in 4 red, 5 blue and 4 green and 2 yellow balls. A kid randomly selects 6 balls without replacement. what is the probability that exactly 3 of the selected balls are blue?

A group of students got the following scores in a test: 6, 9, 12, 15, 18, and 21. Consider sample of size 4 that can be drawn from this population.

How many possible samples of size 4 can be drawn?

List all the possible samples and compute the sample mean.

Sample

Sample Mean

Construct the sampling distribution of the sample mean.

Sample Mean

Frequency

Probability P(x)

Draw a histogram corresponding to the sampling distribution of the sample mean.

Men arrive at a clinic independently and at random, at a constant mean rate of 0.2 per minute. Women

arrive at the same clinic independently and at random, at a constant mean rate of 0.3 per minute.

(i) Find the probability that at least 2 men and at least 3 women arrive at the clinic during a 5-minute

period. [4]

(ii) Find the probability that fewer than 36 people arrive at the clinic during a 1-hour period.

a population size N = 150 has μ = 8 and standard deviation of 5.4. What is the probability that a random size n=20 will have a mean of 9.5 above

On average, 3 students per month cry in happiness. What is the probability that in any given month,

a) exactly 5 students will cry in happiness

b) fewer than 3 students will cry in happiness

c) at least 2 students will cry in happiness

Answers (Decimal Format; 4 decimal places)

What is the equivalent of P (6 ≤ X ≤ 8)?

A) P(X ≤ 8) - P(X ≤ 6)

B) P(X ≤ 8) - P(X ≤ 5)

C) 1- [P(X ≤ 8) + P(X ≤ 6)]

D) P(X ≤ 8) + P(X ≤ 6)

Choose The Right answer

Determine the value c so that each of the following functions can serve as a probability distribution of the discrete random variable X:

(a) f(x) = c(x² + 4), for x = 0, 1, 2, 3;

x=500,n=812,95% confidence