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Fall 2021 1 Introductory calculus course. 1. Show that the function of x is a solution to the differential equation d^2y/dx^2 - a^2y = 0. 2. Note that v = -dy/dx, where dy/dx represents the derivative of y with respect to x. 3. Suppose Y(t) is dependent on a real-valued variable z. Starting with P = Pr(Y = 1) and the Gibbs form P = e^(-E/kT) for some energy E and partition function Z, show that it reduces to the logistic model P = 1/(1 + e^(-z)). 4. Find the range of b such that 0 < b < 1. Find a function for which b has a range of (-1,1) and then express it in terms of the hyperbolic tangent function. Suppose negative N is a random variable with a probability density function (pdf) in black, and positive P is a random variable with a pdf in green as illustrated in the figure. For fixed T in R, the True Positive Rate is defined as TPR(T) = Pr(P > T). What is TPR(T) in terms of p? 6. For fixed T in R, the False Positive Rate is defined as FPR(T) = Pr(N > T). What is FPR(T) in terms of nz?

          Fall 2021
1
Introductory calculus course. 1. Show that the function of x is a solution to the differential equation d^2y/dx^2 - a^2y = 0. 2. Note that v = -dy/dx, where dy/dx represents the derivative of y with respect to x. 3. Suppose Y(t) is dependent on a real-valued variable z. Starting with P = Pr(Y = 1) and the Gibbs form P = e^(-E/kT) for some energy E and partition function Z, show that it reduces to the logistic model P = 1/(1 + e^(-z)). 4. Find the range of b such that 0 < b < 1. Find a function for which b has a range of (-1,1) and then express it in terms of the hyperbolic tangent function. Suppose negative N is a random variable with a probability density function (pdf) in black, and positive P is a random variable with a pdf in green as illustrated in the figure. For fixed T in R, the True Positive Rate is defined as TPR(T) = Pr(P > T). What is TPR(T) in terms of p? 6. For fixed T in R, the False Positive Rate is defined as FPR(T) = Pr(N > T). What is FPR(T) in terms of nz?

need 5 and 6 fall202 1 introductory caeulus courem 1shw tht function of is outio to thediffeential cquntion d a1n 2notkthatv dwher sech the graph of and show that it is n probability deity f 91632

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