# Question: To illustrate the law of large numbers see also Exercise

To illustrate the law of large numbers (see also Exercise 5.54 on page 172), use the normal approximation to the binomial distribution to determine the probabilities that the proportion of heads will be anywhere from 0.49 to 0.51 when a balanced coin is flipped

(a) 100 times;

(b) 1,000 times;

(c) 10,000 times.

(a) 100 times;

(b) 1,000 times;

(c) 10,000 times.

## Answer to relevant Questions

Using the form of the gamma function of Exercise 6.8, we can write And hence Change to polar coordinates to evaluate this double integral, and thus show that (12 ) = v p. In exercise If X is the total we roll with a pair of dice, for which the probability distribution is given on page 63, find the probability distribution of the remainder we get when the values of X are divided by 3. If the probability density of X is given by And Y = X2, find (a) The distribution function of Y; (b) The probability density of Y. If X1 and X2 are independent random variables having binomial distributions with the respective parameters n1 and θ and n2 and θ, show that Y = X1 + X2 has the binomial distribution with the parameters n1 + n2 and θ. Let X and Y be two continuous random variables having the joint probability density Find the joint probability density of Z = X + Y and W = X.Post your question