Which of the following is NOT an assumption or condition that should be met when applying the Central Limit Theorem?
The population standard deviation needs to be known or estimated.
The population data must be normally distributed.
The samples should be independent of each other.
The sample size should be sufficiently large (generally n ≥ 30).
A casino owner wants to ensure that their roulette wheel is fair. How could they apply the Law of Large Numbers to test this?
Use computer simulations to predict the outcomes of roulette spins and compare them to real-world results.
Hire a mathematician to calculate the exact probability of each outcome on the roulette wheel.
Spin the wheel thousands of times and record the results. The distribution of outcomes should be close to the theoretical probabilities.
Spin the wheel a small number of times and analyze the results for any obvious patterns.
The time until a computer system crashes is exponentially distributed with a mean of 2500 hours. What is the probability that the system will crash within the first 1000 hours?
0.3297
0.6703
0.1813
0.5
A machine produces bolts with diameters normally distributed, a mean of 10mm, and a standard deviation of 0.2mm. What percentage of bolts will have a diameter between 9.7mm and 10.3mm?
95.45%
99.73%
86.64%
68.27%
A machine produces widgets with a defect rate of 5%. What is the probability that a batch of 20 widgets will contain exactly 2 defective ones?
0.9975
0.0025
0.1887
0.8113
Which of these is NOT an application of the Law of Large Numbers in data science?
Predicting the outcome of a single coin toss based on past tosses.
Building a spam filter by analyzing the characteristics of millions of emails.
Developing an insurance pricing model based on historical claims data.
Estimating the click-through rate of an online advertisement.
The lifetime of a certain component follows a gamma distribution with a shape parameter of 3 and a rate parameter of 0.2. What is the variance of the component's lifetime?
7.5
45
75
15
You have two standard decks of cards. You draw one card from each deck. What is the probability that both cards are Aces?
1/13
1/169
1/52
1/26
If two random variables X and Y are independent, what can be said about their joint probability distribution?
It is equal to the sum of their marginal distributions.
It cannot be determined from their marginal distributions.
It is always a uniform distribution.
It is equal to the product of their marginal distributions.
A bag contains 5 red balls and 3 blue balls. Two balls are drawn one after another without replacement. What is the probability that the first ball is red and the second ball is blue?
5/28
1/2
3/28
15/56