Monthly sales are independent normal random variables with mean 100 and standard deviation 5. (a) find the probability that exactly 3 of the next 6 months have sales greater than 100. (b) find the probability that the total of the sales in the next 4 months is greater than 420

The probability that exactly 3 of the next 6 months have sales greater than 100 is 0.31 and the probability that the total of the sales in the next 4 months is greater than 420 is 0.42.

Given :

Monthly sales are independent normal random variables with a mean of 100 and a standard deviation of 5.

The probability is given by the formula:

a) The probability that exactly 3 of the next 6 months have sales greater than 100 is given by:

b) The probability that the total of the sales in the next 4 months is greater than 420 is given by:

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Let be the random variable for the value of goods sold in any given month. Also let be the random variable representing the number of times that sales in a given month surpass a value of 100.

follows a normal distribution with mean and standard deviation , while would follow a binomial distribution with some success probability on trials. To find , we use the distribution of :

where is a random variable following the standard normal distribution.

a) The probability that exactly 3 of the next 6 months have sales greater than 100 is then

b) Let be the random variable representing the total value of sales over 4 months, and denote by the random variable for the value of sales during month . Then , and we want to find .

We know the are mutually independent and identically distributed, and we (you probably should, anyway) also know that the sum of i.i.d. normally distributed random variables also follows a normal distribution, whose mean is the sum of the means, and whose standard deviation is the sum of the squares of the standard deviations, of the i.i.d. random variables:

Then the probability we want is

$27.5

step-by-step explanation:

$110.00 divided by 4