Fall 2023 Semester Professor Pandazis Math 241 Test 2 You have one hour to complete the exam. Each question is worth 20 points. Round decimal answers to four decimal places. 1. Prove that b(x; n, p) = b(n - x; n, 1 - p). 2. Let X be the continuous random variable with the following probability density function: f ( x ) = ex, if |x| s 1 otherwise Find the constant c. Find P (x > ;) 3. Suppose we randomly select 4 cards without replacement from an ordinary deck of playing cards. What is the probability of getting exactly 2 red cards? A manufacturer produces light bulbs that are packed into boxes of 100. If quality control studies indicate that 5% of the light bulbs produced are defective, what percentage of the boxes will contain two or more defective light bulbs? Use the Poisson approximation of the binomial distribution to answer this question. Do not find the exact answer. 5 . Suppose X has a continuous uniform distribution over the interval [1.1,5.1]. Find P(1.5