Hi, I am in difficulty when I kept trying to do these R questions.

Could you give me help to solve these R questions?

Thank you so much.

Note: R code and the code output required

Part 1 A radioaCtive object emits particles according to a Poisson process at an average rate of 6.5 particles per second. We observe the object for a total of 7.5 seconds. (a) [1 mark] What is the probability that no more than 50 particles will be emitted during this interval? (b) [1 mark] What is the probability that exactly 46 particles will be emitted during this interval? (c) [2 marks] Suppose it is known that at least 45 particles will be emitted during this interval. What is the probability that no more than 52 particles will be emitted during this interval? Part 2 A manufacturer of ceramic blades estimates that 0.72% of all blades produced are too brittle to use. Suppose we take a random sample of 125 blades and test them for brittleness. We want to nd the probability that at least 3 blades will be too brittle to use. (a) [1 mark] Find the exact probability that at least 3 blades will be too brittle to use. (b) [1 mark] Use an appropriate approximation to nd the apprcuo'nlate probability that at least 3 blades 1will be too brittle to use. Part 3 The fracture toughness (in MPoJ) of a particular steel alloy is lmown to be normally distributed with a mean of 28.8 and a standard deviation of 2.91. We select one sample of this alloy at random and measure its fracture toughness. (a) [1 mark] What is the probability that the fracture toughness will be between 23.2 and 32.7? (b) [1 mark] What is the probability that the fracture toughness will be at least 27.9? (c) [2 marks] Given that the fracture toughness is at least 27, what is the probability that the fracture toughness will be no more than 33.1? Part 4 The purpose of this question is to help you visualize the normal approximation to the binomial distribution which have seen in Set 16. (a) [1 mark] Let X ~ Mamion = 75.1: : 0.43]. Create a vector called simulation.data which contains a simulation for 3900 values for X. (Le. Simulate 3900 experiments, each being binomial with n = 75 and p = 0.43.} Provide a copy of the R command which you used to create this vector. You do not need to oopy the 3900 values you generated. (b) [2 marks] Create a histogram of simulationdata and copy it and your line of R. code into your assignment. Your histogram should have an appropriate title and an appropriate label on the :r-axis. Comment on the shape of the histogram. [We are looking for a single phrase here to describe the histogram. It should be a shape we've discussed recently.) (c) [2 marks] Calculate the sample mean of simulation.data. Copy the command used, and the output. How close is your sample mean to what you 1would expect? (Hint: We have discussed the expected value of the sample mean T. We have also discussed the expected value of a binomial random variable X.)