3 / 12 = 1 / 4 = 0.25 . pg. 137. Correct answer is C.

arithmetic mean : m = 850 / 5 = 170 ; s = ((60² + 20² + 20² + 10² + 55²) / 4)^1/2 = ((3.600 + 400 + 400 + 100 + 3.025) / 4)^1/2 = (7.525 / 4)^1/2. pg. 111. Correct answer is B.

Pearson’s coefficient of skewness (PCS) can take values between -3 and 3. A negative value near -3 shows that the distribution is considerably left skewed and a positive value near 3 shows that the distribution is considerably left-skewed. If the PCS is near zero, this indicates that the distribution is symmetric because in this case the mean, the median, and the mode are similar.

The range of a data set, shown as R, is the difference between the largest and smallest values and calculated as follows: R = Largest Value - Smallest Value

R= 12345 -3600 = 8745

Exponential distribution is another most significant and extensively used continuous probability distribution. Exponential random variable is frequently used to model the time interval between two events. Exponential random variable is defined with a parameter ? and it’s represented as X~Exponential (?). In that sense the exponential random variable X defines the time interval between two consecutive events of a Poisson process with a mean of µ = ?. Here ? parameter defines the number of events is a certain time period. Therefore, it’s essential to use consistent time units in the determination of probabilities, mean and variance with the exponential random variable X. The mean (µ) and variance (?2) for exponential random variable X~Exponential (?) with parameter ? can be calculated from the following formulas,
E(x) = µ = 1/?
V (x) =?2 = 1/?2

Hence from above given formulas it’s clear that the mean and the standard deviation of the exponential distribution are equal.

P(A) * P(B) + P(not-A) * P(C) = 0.04 * 0.03 + 0.96 * 0.02 = 0.0012 + 0.0192 = 0.0204. pg. 138. Correct answer is D.

=  19,72

The property of "If x1 ≤ x2 then F (x1) ≤ F (x2)"  is not valid related to probability density function f (x) for normal distribution because it is the property of Cumulative Density Function F(x) of a continuous random variable X.

The mean can be calculated by multiplying the weight with their frequencies and summing them up.

Thus average (mean) weight=60*(30/100)+70*(40/100)+80*(20/100)+90*(10/100)=18+28+16+9=71

25 / (25 + 60) = 25 / 85 = 5 / 17. pg. 141. Correct answer is E.

Bayes’ Theorem is an important application of conditional probability. This theorem enables us to compute a particular conditional probability. Using Bayes’ Theorem, we can find the probability that a defective item is produced by a particular machine, 

Normal distribution provides basis for the statistical inference. Normal distribution is a symmetric distribution where the random variable values are uniformly scattered around the mean. Population mean, µ and standard deviation ? parameters determine the shape of the normal distribution function. Uniform distribution is different than normal distribution.

First property assures that probability density function of random variable x obtain the non-negative values at all times. From the shape of the probability density function curve it’s obvious that pdf, f (x) decreases as the random variable value goes away from the mean, µ. Likewise, probability density function f (x) increases as the random variable value gets closer to the mean, µ. Second property identifies that the area under the probability density function f (x) always equivalent to 1 in the definition interval of the random variable X. Third property suggests that normal distribution curve has a similar shape on both sides of the mean x=µ. That property also proposes the fact that, P (X < µ ) = P (X > µ ) =0.5. Fourth property indicates that the tails of the probability function goes to infinity and at no time crosses or touches the x axis.

Mean = (0 * 1/8)  + (1 * 3/8)  +  (2 * 3/8)  + (3 * 1/8) =10/8=1.25

Sometimes it will be important to consider events that do not occur at the same time. That is the events whose intersection is the null event (empty set). If A and B are any two events then they are said to be mutually exclusive if  A ∩ B = 0. The answer is B.

Scatter plot is used to investigate the relationship between two variables. They are also very helpful indicating the minimum, maximum or outliers of the variables. One of the reasons that the scatter plots may be drawn is that scatter plot gives a good indication about the correlation between two variables.