Problem 1:
Suppose that two fair dice are tossed and the random variable observed-say, x - is the
sum of the two up faces. Describe the sample space of this experiment, and determine the
probability distribution of x.
a) Calculate the probability of rolling a 2
b) Calculate the probability of rolling a 3
c) Calculate the probability of rolling a 4
d) Find the mean and variance of the random variable
e) Find the probability that the result x is comprised greater than 8 .
Problem 2:
A quality characteristic of a product is normally distributed with mean mu and standard
deviation sigma =1. Specifications on the characteristic are 6<=x<=8. A unit that falls within
specifications on this quality characteristic results in a profit of C_(0). However, if x<6, the
profit is C_(1), whereas if x>8, the profit is C_(2). Find the value of mu that maximizes the
expected profit.
Problem 3:
The billing department of a major credit card company attempts to control errors (clerical,
data transmission, etc.) on customers' bills. Suppose that errors occur according to a Poisson
distribution with parameter lambda =0.01. What is the probability that a customer's bill selected
at random will contain one error?
Problem 4:
The random variables x and Y are normally distributed with parameters (mu =20,sigma =2.5)
and (mu =5,sigma =1.5), respectively. Z is another random variables defined as:
Z=x+2Y.
Find the probability distribution function associated with Z.
What is the probability that Z will take values between 15 and 30 ?
Problem 1:
Suppose that two fair dice are tossed and the random variable observed-say, r--is the sum of the two up faces. Describe the sample space of this experiment. and determine the probability distribution of x.
a) Calculate the probability of rolling a 2 b) Calculate the probability of rolling a 3 c) Calculate the probability of rolling a 4 d) Find the mean and variance of the random variable e Find the probability that the result is comprised greater than 8
Problem 2:
A quality characteristic of a product is normally distributed with mean and standard deviation o = 1. Specifications on the characteristic are 6 < x < 8. A unit that falls within specifications on this quality characteristic results in a profit of Co. However, if < 6, the profit is Ci, whereas if > 8, the profit is C2. Find the value of that maximizes the expected profit.
Problem 3:
The billing department of a major credit card company attempts to control errors (clerical. data transmission, etc.) on customers' bills. Suppose that errors occur according to a Poisson distribution with parameter A = 0.0l. What is the probability that a customer's bill selected at random will contain one error?
Problem 4:
The random variables X and Y are normally distributed with parameters ( = 20, = 2.5 and ( = 5, = 1.5), respectively. Z is another random variables defined as:
Z = X + 2Y.
l. Find the probability distribution function associated with Z
2. What is the probability that Z will take values between 15 and 30?