91Ó°ÊÓ

Mean as center of Gravity. Let X be a discrete random variable with a finite number of number of possible values say $x_1,x_2.....x_m$For convenience, set $P_K=P(X=x_k),$for K = 1, 2.....m . Think of a horizontal axis as a seesaw and each $P_K$as a mass placed at point $x_k$on the seesaw. The center of gravity of those masses is defined to be the point c on the horizontal axis at which a fulcrum could be placed to balance the seesaw.

Relative to the center of gravity , The torque acting on the seesaw by the mass $P_K$is proportional to the product of that mass with the signed distance of the point $X_k$From c That is , to ($(x_k-c)$. $P_k$show that the center of gravity equal the mean of the random variable X ( hint: To balance, the total torque acting on the seesaw must be 0)

Under what three conditions are repeated trials of an experiment called Bernoulli trials?

Explain the significance of binomial coefficients with respect to Bernoulli trials.

Discuss the pros and cons of binomial probability tables.

Suppose that a simple random sample is taken from a finite population in which each member is classified as either having or not having a specified attribute. Fill in the following blanks.

(a) If sampling is with replacement, the probability distribution of the number of members sampled that have the specified attribute is a distribution.

(b) If sampling is without replacement, the probability distribution of the number of members sampled that have the specified attribute is a distribution.

(c) If sampling is without replacement and the sample size does not exceed % of the population size, the probability distribution of the number of members sampled that have the specified attribute can be approximated by a distribution.

Give two examples of Bernoulli trials other than those presented in the text.

In Exercises 5.16-5.26, express your probability answers as a decimal rounded to three places.

Preeclampsia. Preeclampsia is a medical condition characterized by high blood pressure and protein in the urine of a pregnant woman. It is a serious condition that can be life-threatening to the mother and child. In the article "Women's Experiences Preeclampsia: Australian Action on Preeclampsia Survey of Wom and Their Confidants" (Journal of Pregnancy,Vol. 2011, Issue 1, Article ID 375653), C. East et al. examined the experiences of 68 women with preeclampsia. The following table provides a frequency distribution of instances of prenatal or infant death for infants of women with preeclampsia.

Suppose that one of these women with preeclampsia is randomly selected. Find the probability that the child of the woman selected

(a) died.

(b). died one week to six months after birth.

(c). lived at least six weeks.

According to the Daily Racing Farm, the probability is about \(0.67\) that the favorite in a horse race will finish in the money (first, second or third place). In the next five races, what is the probability that the favorite finishes in the money.

a. exactly twice?

b. exactly four times

c. at least four times?

d. between two and four times, inclusive?

e. Determine the probability distribution of the random variable \(X\), the number of times the favorite finishes in the money in the next five races.

f. Identify the probability distribution of \(X\) as right skewed, symmetric or left skewed without consulting its probability distribution or drawing its probability histogram.

g. Draw a probability histogram for \(X\).

h. Use your answer from part (c) and definitions \(5.9\) and \(5.10\) on pages \(227\) and \(229\) respectively to obtain the mean and standard deviation of the random variable \(X\).

i. Use formula \(5.5\) on page \(239\) to obtain the mean and standard deviation of the random variable \(X\).

j. Interpret your answer for the mean in words.

Traffic Fatalities and Intoxication. The National Safety Council publishes information about automobile accidents in Accident Facts. According to that document, the probability is 0.40 that a traffic fatality will involve an intoxicated or alcohol-impaired driver or nonoccupant. In eight traffic fatalities, find the probability that the number,Y, that involve an intoxicated or alcohol-impaired driver or nonoccupant is

(a) exactly three; at least three; at most three.

(b) between two and four, inclusive.

(c) Find and interpret the mean of the random variable Y.

(d) Obtain the standard deviation of Y.

In Exercises 5.16-5.26, express your probability answers as a decimal rounded to three places.

Cardiovascular Hospitalizations. From the Florida State Center for Health Statistics report Women and Cardiovascular Disease Hospitalization, we obtained the following table showing the number of female hospitalizations for cardiovascular disease, by age group, during one year.

One of these case records is selected at random. Find the probability that the woman was

(a) in her 50s.

(b) less than 50 years old.

(c) between 40 and 69 years old, inclusive.

(d) 70 years old or older.