The time (in minutes) between customer arrivals at a bank is exponentially distributed with an average of 5 minutes. What is the probability a customer arrives within 3 minutes of the previous customer?
0.368
0.451
0.632
0.549
In a large dataset of customer purchase amounts, the average purchase is $50 with a standard deviation of $20. If we take 100 random samples of size 25 from this dataset, what will be the standard deviation of the distribution of these sample means?
$4
$2
$10
$20
A bag contains 5 red marbles and 5 blue marbles. You draw two marbles from the bag without replacement. What is the probability that both marbles are red?
2/9
1/4
1/2
1/9
In a data science context, joint probability distributions are often visualized using:
Bar charts
Scatter plots
Heatmaps
Pie charts
The lifetime of a certain battery follows a gamma distribution with a shape parameter of 2 and a scale parameter of 500 hours. What is the average lifetime of this battery?
1000 hours
500 hours
250 hours
2000 hours
A company wants to analyze its website traffic. The time a user spends on the website follows a gamma distribution. Which of these is NOT a suitable use of the gamma distribution in this scenario?
Modeling the total time users spend on specific pages.
Analyzing the distribution of session durations.
Predicting the time until a user makes a purchase.
Estimating the probability of a user clicking on an advertisement.
You roll two six-sided dice. Are the events "getting a sum of 7" and "getting doubles" independent events?
Yes
No
You're analyzing the average height of trees in a forest. You take multiple samples of 50 trees each. According to the Central Limit Theorem, what can you infer about the distribution of the sample means of these tree heights?
The distribution of sample means will be identical to the distribution of individual tree heights.
The Central Limit Theorem cannot be applied to this situation.
The distribution of sample means will be approximately normal.
The distribution of sample means will be skewed right.
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?
15
7.5
45
75
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.