This implies an assumption that the probability of each sample is independent from the others. Select one: A. True B. False
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This implies an assumption that the probability of each sample is independent from the others.
Select one:
A. True
B. False
Step by step
Solved in 2 steps with 2 images
- 1. In a two-class problem, the likelihood ratio is p(x|C₁) p(x|C₂) Write the discriminant function in terms of the likelihood ratio.Solve In R programmning language: Calculate the probability for each of the following events: (a) A standard normally distributed variable is less than -2.5. (b) A normally distributed variable with mean 35 and standard deviation 6 is larger than 42 but less than 45. (c) A normally distributed variable with mean 35 and standard deviation 6 is larger than 40 but less than 41. (d) X < 0.9 when X has the standard uniform distribution (min=0, max=1). (e) 1 < X < 3 in the exp distribution with rate λ = 2.If a problem has 12 boolean variables, how many full joint probabilities would you need to add up to find P(a v b)
- For a given joint probability distribution f (x, y) where 0 < x < 4 and 0 < y < 5, which of the following equation provides the marginal probability of y? O f f (x, y) dy O S f (x, y) dx S f (x, y) dx O sf f (x, y) dæFor logistic regression, the gradient of the cost function is given by J(0) = (i) E (he (x) – y')x;). Write down mathematical expression(s) for the correct m gradient descent update for logistic regression with a learning rate of a. (In the expression, he(x^) should be replaced by the sigmoid function.)Consider the same house rent prediction problem where you are supposed to predict price of a house based on just its area. Suppose you have n samples with their respective areas, x(¹), x(²),...,x(n), their true house rents y(¹), y(2),..., y(n). Let's say, you train a linear regres- sor that predicts f(x)) = 0 + 0₁x). The parameters, and 0₁ are scalars and are learned by minimizing mean-squared-error loss with L1-regularization through gradient descent with a learning rate a and the regularization strength constant A. Answer the following questions. 1. Express the loss function(L) in terms of x(i),y(i), n, 00, 01, X. 2. Compute L 200 ƏL 3. Compute 20₁ 4. Write update rules for 0o and 0₁ Hint: d|w| dw undefined -1 w>0 w=0 w <0
- In class, we have defined the hypothesis for univariate linear regression as he(x) = 00 + 0₁x and the mean-square cost function as 2 J(00,01) Σ (ho(x²) - y(i)) ² 2m i=1 We define the following terms: m `x), y= 10. ;Σy"), xy = - ² x(1) y(i) x²: (x(1)) ² m a. Show that the optimal values of the learning parameters 00 and 0₁ are given by: 00 = y-x xy−xy x²-x² 01 = ху-ху x²-x² b. What is the advantage of expressing the learning parameters in this format? =From the question 1, given P(D|M) = P(M|D)P(D) p(M) solve the following a. Draw the probability tree for the situation.The sample space of a random experiment is (a, b, c, d, e, f, and each outcome is equally likely. A random variable is defined as follows: outcome X с d TEISEER 1.5 1.5 a b 0 2 Determine the probability mass function of X. Use the proba- bility mass function to determine the following probabilities: (a) P(X= 1.5) (b) P(0.5 3) (d) P(0 ≤X<2) (e) P(X=0 or X = 2)
- Consider the same house rent prediction problem where you are supposed to predict price of a house based on just its area. Suppose you have n samples with their respective areas, x(1), x(2), ... , x(n), their true house rents y(1), y(2),..., y(n). Let's say, you train a linear regres- sor that predicts f(x()) = 00 + 01x(e). The parameters 6o and 0, are scalars and are learned by minimizing mean-squared-error loss with L2-regularization through gradient descent with a learning rate a and the regularization strength constant A. Answer the following questions. 1. Express the loss function(L) in terms of x), y@), n, 0, 01, A. 2. Compute L 3. Compute 4. Write update rules for 6, and O1A particular telephone number is used to receive both voice calls and fax messages. Suppose that 20% of the incoming calls involve fax messages, and consider a sample of 20 incoming calls. (Round your answers to three decimal places.) (a) What is the probability that at most 6 of the calls involve a fax message?(b) What is the probability that exactly 6 of the calls involve a fax message?(c) What is the probability that at least 6 of the calls involve a fax message?(d) What is the probability that more than 6 of the calls involve a fax message?Solve in R programming language: 5) Use R to calculate and simulate with the exponential distribution as follows. (a) For an exponential random variable X with λ = 4, simulate 1000 independent exponential random variables by using the R function rexp(n, λ). Calculate the mean and variance of this sample. (b) Compare the empirical results from part (a) with the distribution mean 1/λ and the distribution standard deviation 1/λ.