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Suppose that you are given the distribution functions of two random variables X and Y. If Cov(X,Y)=0, are X and Y independent? Answer True/False below and provide a supporting reason on your scratch paper. An answer without a reason will receive 0. If this is not sufficient to determine whether they are independent, choose False and explain why.

Compute the conditional probability P(X>1.5∣X>1). Provide three digits after the decimal point.

Let X be a continuous random variable with the following probability density function:

What is the probability that a randomly selected student knows at least one of the three languages? Compute three digits after the decimal point.

A venture capital firm evaluates thousands of startup companies each year. Historically, only 1 out of every 1,000 startups turns out to be successful. The firm uses an initial evaluation test to assess potential. The evaluation test is imperfect: If a startup will be successful, the test returns a positive evaluation with probability 0.8 If a startup will not be successful, the test returns a positive evaluation with 0.05 Show your work on the scratch paper answering Q4-Q6.

What is the probability that a student who knows R also knows at least one of the other two languages? Compute three digits after the decimal point.

If a startup receives a negative evaluation, what is the probability that it will not be successful? Provide six digits after the decimal point.

A factory produces metal rods whose weights (in pounds) are independent and identically distributed random variables with mean μ = 5 and standard deviation σ = 0.2. The rods are packaged in boxes of 30. Show your work on the scratch paper answering Q16-Q17.

A graduate-level statistics class surveyed 120 students on their knowledge of three software tools: R, Python, and MATLAB. 65 students know R 75 students know Python 50 students know MATLAB 30 students know both R and Python 25 students know both R and MATLAB 35 students know both Python and MATLAB 15 students know all three languages Show your work on the scratch paper when answering Q1-Q3.

What is the advantage of the code on the top compared to the one on the bottom. (Note: BalanceException is a checked exception)   int withdraw(int amount) {   if (amount > balance)       return -1;   else {       balance -= amount;        return 0;   } }   void withdraw(int amount) throws BalanceException{  if (amount > balance){        throw new BalanceException();   }   balance -= amount; }