How is the marginal probability distribution of X obtained from the joint probability distribution of X and Y?
By multiplying the joint probabilities by the marginal probabilities of Y
By dividing the joint probabilities by the marginal probabilities of Y
By subtracting the joint probabilities from 1
By summing the joint probabilities over all possible values of Y
The time it takes for a certain type of light bulb to burn out is exponentially distributed with a mean of 500 hours. What is the probability that a randomly selected bulb will last longer than 1000 hours?
0.1353
0.6065
0.3679
0.5
If the conditional probability P(Y=y | X=x) is equal to the marginal probability P(Y=y) for all values of x and y, what does this imply about the relationship between X and Y?
X and Y are independent.
X and Y are dependent.
X and Y are uniformly distributed.
X and Y are mutually exclusive.
The amount of time customers spend on a website follows a gamma distribution with a shape parameter of 2 and a scale parameter of 5 minutes. What is the expected amount of time a customer will spend on the website?
5 minutes
2.5 minutes
20 minutes
10 minutes
The weight of a bag of chips is normally distributed with a mean of 250 grams and a standard deviation of 5 grams. If a bag is selected at random, what is the probability that it weighs less than 240 grams?
0.9772
0.4772
0.0228
What is a key difference between the Weak Law of Large Numbers and the Strong Law of Large Numbers?
The Weak Law considers a finite number of trials, while the Strong Law considers an infinite number of trials.
The Weak Law applies only to discrete distributions, while the Strong Law applies to both discrete and continuous distributions.
The Weak Law is used in hypothesis testing, while the Strong Law is used in confidence interval estimation.
The Weak Law deals with the convergence in probability, while the Strong Law deals with almost sure convergence.
A fair coin is tossed three times. What is the probability of getting at least two heads?
1/8
3/8
7/8
1/2
You are simulating a random process with a known theoretical mean. After many simulations, you find that the average of your simulated results is significantly different from the expected theoretical mean. What is the MOST likely explanation for this discrepancy?
You have not run enough simulations for the Law of Large Numbers to take effect.
The Law of Large Numbers does not apply to simulations.
The theoretical mean is incorrect.
There is an error in your simulation code or methodology.
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?
86.64%
68.27%
99.73%
95.45%
The heights of adult males in a certain population are normally distributed with a mean of 175 cm and a standard deviation of 7 cm. What percentage of men are taller than 189 cm?
5%
32%
2.5%
16%