Descriptive Statistics
Tutorial Sheet 6
Hint: Using the results regarding the mean and variance of a variable from a standard
distribution (see lecture notes) will be useful.
1. 192 numbers rounded to the nearest integer are added together. Assuming that the
rounding errors are independent random variables from the uniform distribution on
[-0.5, 0.5], estimate the probability that the total error in the calculation of the sum is
a) greater than 5
b) between -3 and 2.
2. Estimate the probability that the sum of 100 independent exponential random
variables with expected value 2 is between 190 and 220.
3. a) A coin is tossed 100 times. Using an appropriate approximation, estimate the
probability that
i) exactly 46 heads are thrown
ii) between 48 and 53 heads (inclusively) are thrown
4. There are 1000 balls in a bag. 400 are white and the remaining balls are black. 200
balls are chosen with replacement.
i) Using an appropriate approximation, estimate the probability that
a) more than 85 white balls are chosen
b) between 72 and 86 (inclusively) white balls are chosen.
ii) Repeat the calculations from i) under the assumption that the balls are chosen
without replacement.
Hint: Use the formulas for the expected value and variance for the binomial distribution and
hypergeometric distribution, as appropriate.