Compute the expected value, E, in hours. Compute three digits after the decimal point.

Compute the variance of the number of premium coffees purchased per customer. 

The probability that the sample average is greater than $210 can be written as

Given that a student spends more than 1 hour in the cafe, find the probability they spend more than 3 hours. Compute three digits after the decimal point.

A startup business school cafe tracks the number of premium coffees a randomly chosen customer buys in one visit. The random variable XXX denotes this number. Based on past data, the probability mass function of X is:  P(X=x)={0.1,x=0(no premium coffee),0.2,x=1(one premium coffee),0.3,x=2(two premium coffees),0.4,x=3(three premium coffees),0,otherwise.P(X=x) = \begin{cases} 0.1, & x=0 \quad (\text{no premium coffee}), \\ 0.2, & x=1 \quad (\text{one premium coffee}), \\ 0.3, & x=2 \quad (\text{two premium coffees}), \\ 0.4, & x=3 \quad (\text{three premium coffees}), \\ 0, & \text{otherwise}. \end{cases}

Compute the standard deviation of X in hours. Compute three digits after the decimal point.

Using ________________________________________ approach, probabilities can only come from experiments.

The figure below shows two overlapping probability distributions: The left distribution (I) corresponds to impostor attempts, the right distribution (A) corresponds to authentic users, and a decision threshold is used to decide whether to accept or reject a person. Four possible outcomes exist depending on which side of the threshold the sample falls (A-D each letter is marked in a distinct region of the figure). Match each outcome below with the correct region: False Positive (Type II Error) (5 points) Correct Rejection (True Negative) (5 points) False Negative (Type I Error) (5 points) Correct Acceptance (True Positive) (5 points)

In the two-category case, if the class-conditional densities are Gaussian and the covariance matrix for each class is arbitrary, the decision surfaces are __________________________________________ (5 points).

An experiment whose result is not certain in advance is _______________________________________________________.