Probability work | Mathematics homework help

QUESTION 4

1.      Exercise 4.0a 

You are visiting the rainforest, but unfortunately your insect repellent has run out. As a result, at each second, a mosquito lands on your neck with probability 0.5. If a mosquito lands, it will bite you with probability 0.2, and it will never bother you with probability 0.8, independently of other mosquitoes. What is the probability of being bit for the 1st time in the 5th second?

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2 points   

QUESTION 5

1.      Exercise 4.0b

You toss n = 10 fair coins, each showing heads with probability p₁ = 0.5, independently of the other tosses. Each coin that shows tails is then tossed again, once. After following this process, what is the probability that exactly k = 6 coins show heads?

Hint: first calculate p₂,the probability that a single coin shows heads (after following the 2-step process), and then make use of the binomial PMF.

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2 points   

QUESTION 6

1.      Exercise 5.3a

Let a continuous random variable X be given that takes values in [0, 1], and whose distribution function F satisfies

F(x) = 2x² − x⁴    for 0 ≤ x ≤ 1

a. Compute P (1/4 ≤ X ≤ 3/4 )

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QUESTION 10

1.      Exercise 5.14

Determine the 10th percentile of a standard normal distribution