Probability And Statistics

We are given the first three raw moments about an arbitrary point . Let's denote them by . So, , , and .

Step 1: Calculate the Mean. The mean of the distribution, , is the first moment about the origin. We can find it using the first moment about point A. The formula is: Mean .

Step 2: Calculate the Variance. The variance, , is the second central moment (the second moment about the mean), denoted by . The formula to convert the second raw moment to the second central moment is: . So, the mean of the distribution is 3 and the variance is 15.

Step 1: The mean of the data is given as 5.
Step 2: The variance is given. We know that the variance is the average of the squared deviations from the mean. Thus:
Where and .
Step 3: We know the following three observations: 1, 2, and 6. So, the sum of squared deviations for these is:
Step 4: Since the variance is 124, we have the equation for the variance as:
Step 5: To find the mean deviation, we calculate the mean of the absolute deviations of all the data from the mean . The formula for mean deviation is:
The mean deviation for the known values is:
Now for the remaining two unknown observations, we can deduce that the total mean deviation for all the observations is approximately 2.8. Thus, the correct answer is (c) 2.8.

The variance is the average of the squared deviations from the mean, so it is expressed in square units of the units of observation. Therefore, the statement that the variance is expressed in the same units as the units of observation is incorrect. The correct answer is option (b).

Step 1: Standard Deviation and Mean Deviation.
The standard deviation is always greater than or equal to the mean deviation. This is because the sum of the absolute deviations (mean deviation) is less than or equal to the square root of the sum of squared deviations (standard deviation).

Step 2: The Units of Variance.
Variance is calculated as the average of squared deviations, so its units are the square of the units of observation, unlike the standard deviation, which is in the same units as the data.

Step 3: Scale Effect on Standard Deviation.
The value of the standard deviation changes with scale. If we scale the data by a factor, the standard deviation is also scaled by the same factor. Thus, the correct answer is option (b).

Step 1: Understand the Poisson distribution.
The Poisson distribution for the number of events occurring in a given interval of time (here, two weeks) is given by: where is the average number of events in the interval (here, 2 accidents per week, so for two weeks).

Step 2: Apply the formula for .
We are given that the probability is , so we substitute the values into the Poisson formula: Thus, comparing this to , we find that .

We are given that the mean score and the standard deviation . To find the probability that a student scored between 450 and 500, we first convert the scores to standard normal form using the Z-score formula: For : For : From the standard normal table, the cumulative probability for is 0.3085, and for is 0.5. The probability that a student scored between 450 and 500 is the difference between these two values: Thus, the probability is approximately 0.191.