Practice: Assignment 0 - (Non- Graded)

A medical company touts its new test for a certain genetic disorder. The false negative rate is small: if you have the disorder, the probability that the test returns a positive result is 0.999. The false positive rate is also small: if you do not have the disorder, the probability that the test returns a positive result is only 0.005. Assume that 2% of the population has the disorder. If a person chosen uniformly from the population is tested and the result comes back positive, what is the probability that the person has the disorder?

There are n bins of which the k-th bin contains k − 1 blue balls and n − k red balls. You pick a bin at random and remove two balls at random without replacement. Find the probability that: • the second ball is red; • the second ball is red, given that the first is red.

In an experiment, n coins are tossed, with each one showing up heads with probability p independently of the others. Each of the coins which shows up heads is then tossed again. What is the probability of observing 5 heads in the second round of tosses, if we toss 15 coins in the first round and p = 0.4? (Hint: First find the mass function of the number of heads observed in the second round.)

Consider two random variables X and Y having joint density function \displaystyle f (x, y) = 2e^{−x−y} , 0 < x < y < ∞ . Are X and Y independent? Find the covariance of X and Y .

An airline knows that 5 percent of the people making reservations on a certain flight will not show up. Consequently, their policy is to sell 52 tickets for a flight that can hold only 50 passengers. What is the probability that there will be a seat available for every passenger who shows up?

Let X have mass function \displaystyle f(x) =\begin{cases} \{x(x+1) \}^{-1} & if \,\, x = 1,2,..., \\ 0 & otherwise, \end{cases} and let \displaystyle α ∈ \mathbb{R}. For what values of \displaystyle α is it the case that \displaystyle \mathbb{E}(X^α ) < ∞ ?

Is the following a Cumulative Distribution Function? \displaystyle F(x) =\begin{cases} e^{-1/x} & x > 0 \\ 0 & otherwise, \end{cases} If so, give the corresponding density function. If not, mention why it is not a distribution function.

Can the value of a probability density function be greater than one? What about the cumu- lative distribution function?

You are given a biased coin with probability of seeing a head is p = 0.6 and probability of seeing a tail is q = 0.4. Suppose you toss the coin 10 times, what is the probability of you getting the head at most 2 times? Also, what is the probability of you getting the head for the first time on your fourth attempt?

Given a bag containing 6 red balls, 4 blue balls and 7 green balls, what is the probability that in 5 trials, at least 3 red balls are drawn from the bag?

In the experiment from the previous question, what is the probability of picking a red ball for the first time on the fourth attempt?

Suppose we have a set of N balls that we repeatedly choose from. We have a preferred ball ’B’, but instead of selecting our preference all the time, with a small probability \displaystyle \epsilon, we instead select a ball at random from among all balls N with equal probability. What is the probability that our preferred ball is selected on any single try?

The soft-max function takes a vector of real numbers u and normalizes it into a discrete probability distribution. \displaystyle Prob(u_i) = \frac{exp \, (u_i / τ)}{ \sum _{j=1}^n exp(u_j / τ)} Suppose we sample a single index value according to the softmax probability distribution defined over the vector u. That is, with \displaystyle P rob(u_i ) we select index \displaystyle i. For what value of τ are we most likely to select the index with the highest value \displaystyle u_i ?

Suppose we sample a set of numbers \displaystyle \{x_1 , x_2 , ..., x_n \}. Which of the following recursive formulas represents a weighted average where newer samples are weighted higher than older samples as \displaystyle n becomes large. (\displaystyle \overline{x}_n represents the weighted average of the n samples)

You are given a dataset of inputs X= [[1., 0.], [0.,1.]] and targets Y = [0, 1]. Using basis vectors [1, x_1 , x_2 ], and the prediction function f (x) = w^T x, you need to compute a single epoch of gradient descent training, to minimize Mean Squared Error (MSE). Use learning rate η = 1.0, along with the initial weight setting w = [1., 1., 1.]. \displaystyle MSE(X,Y) = \frac{1}{2n} \sum_{i=1}^{n} \big(y_i - f(x_i) \big)^2 The weight vector w after a single epoch of training is: