1. Binomial Random Variable: A company buys 500 laptops for their employees. For each laptop, the probability the laptop doesn't work when purchased is p = 0.01; the probability the laptop works when purchased is 1 - p = 0.99.
What is the probability that all of the 500 laptops work when purchased, PX(0)?
2. Poisson Random Variable The Poisson distribution (or PMF) is the limiting case of the binomial distribution (or PMF) as n → ∞, assuming that the expected value E[X] stays constant (so success or failure, depending on your definition, becomes a rare event). A good rule of thumb for when the Poisson PMF is a good approximation to the binomial PMF is for n ≥ 20 and p ≤ 0.05, with np ≤ 10.
The mean or expected value E[X] of a Poisson random variable is E[X] = α. The mean of a binomial random variable is np. To use a Poisson PMF to approximate a binomial PMF with large n, the expected values must be the same. Therefore, E[X] = α = np.
(a) Could a Poisson PMF be used as a good approximation to the previous problem, which used a binomial PMF? Explain why or why not.
(b) If we use a Poisson PMF as an approximation to the binomial PMF PX(x) for X in the previous problem, what is the expected value, E[X]? What is α?
(c) Write the PMF PX (x) when using the Poisson PFM.
(d) What is the probability that no laptops work when purchased, PX(0)? Be sure to use the Poisson PMF.
(e) Is your result for PX (0) with the Poisson PMF close to your result for PX (0) using the binomial PMF?
3. Variance: Consider the 7-bit ASCII code. Assume that every bit is equally likely a 0 or a 1. Let X be a random variable that is the number of 1's in a 7-bit ASCII codeword. We can represent the PMF of X as a binomial PMF, so X is a binomial random variable. The probability of success, p, is the probability that a bit is 1.
(a) What is p? What is n?
(b) Write the PMF of X, PX (x). Include numeric values for n and p.
(c) The expected value of a binomial random variable X is E[X] = np, from Theorem 2.7(a) in the textbook. Find the numeric value of E[X] with the n and p values found in this problem.
(d) Find the variance of X, σ2X.
(e) Find the probability that X is within one standard deviation of the mean, P(µX - σX ≤ X ≤ µX + σX ).