Download Probability and Random Variables in Engineering Statistics and more Schemes and Mind Maps Economics in PDF only on Docsity!
Ghulam Ishaq Khan Institute of Engineering Sciences and Technology (GIKI) Faculty of Engineering Sciences (FES) ES 202 Spring 2023 Practice Assignment 2 CLO 2 : Calculate probability mass/density function parameters, moments and functions of random variables. Q 1. A tobacco company produces blends of tobacco, with each blend containing various proportions of Turkish, domestic, and other tobaccos. The proportions of Turkish and domestic in a blend are random variables with joint density function (X = Turkish and Y = domestic)
xy x y x y
f x y
otherwise
^ ^ ^ +^
a. Find the probability that in a given box the Turkish tobacco accounts for over half the blend. b. Find the marginal density function for the proportion of the domestic tobacco. c. Find the probability that the proportion of Turkish tobacco is less than 1/8 if it is known that the blend contains 3/4 domestic tobacco. Q 2. An insurance company offers its policyholders a number of different premium payment options. For a randomly selected policyholder, let X be the number of months between successive payments. The cumulative distribution function of X is
0, if 1,
0.4, if 1 3,
0.6, if 3 5,
0.8, if 5 7,
1.0, if 7.
x
x
F x x
x
x
^ =
a. What is the probability mass function of X? b. Compute P(4 < X ≤ 7).
Q 3. Suppose that (^ )^ / 3 ,^0
x
^ f^ x^ =^ c^ x is the probability function for a
random variable X. a. Determine c. b. Find the cumulative distribution function. c. Graph the probability mass function and the cumulative distribution function.
d. Find P^ (^2 ^ X ^5 ).
e. Find P^ (^ X^ ^3 ).
Q 4. Suppose that
2
x
cxe x
f x
otherwise
−
is the density function for a random variable X. a. Determine c. b. Find the cumulative distribution function. c. Graph the probability density function and the cumulative distribution function.
d. Find P^ (^ X^ ^1 ).
e. Find P^ (^2 ^ X ^3 ).
Q 5. The shelf life of a product is a random variable that is related to consumer acceptance. It turns out that the shelf life Y in days of a certain type of bakery product has a density function
y
e y
f y
otherwise
particular system containing three components, the probabilities of meeting specifications for components 1, 2, and 3, respectively, are 0.95, 0.99, and 0.92. What is the probability that the entire system works? Q 9. One type of system that is employed in engineering work is a group of parallel components or a parallel system. In this more conservative approach, the probability that the system operates is larger than the probability that any component operates. The system fails only when all components fail. Consider a situation in which there are 4 independent components in a parallel system with probability of operation given by Component 1: 0.95; Component 2: 0.94; Component 3: 0.90; Component 4: 0.97. What is the probability that the system does not fail? Q 10. Consider a system of components in which there are 5 independent components, each of which possesses an operational probability of 0.92. The system does have a redundancy built in such that it does not fail if 3 out of the 5 components are operational. What is the probability that the total system is operational?
