Problem 1 Let X be a Poisson random variable with parameter 1 > 0, i.e. Ak P{X = k) = e-¹. k! Show that EX = A. Hint: Use that /k! = e^. -k=0
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- A poisson random variables has f(x,3)= 3x e-3÷x! ,x= 0,1.......,∞. find the probabilities for X=0 1 2 3 4 and also find mean and variance from f(x,3).?A random variable X is uniform from 4 to 8. A Gaussian random variable Y has mean of 10. Approximate the variance of Y if only 2.5% of its elements is a subset of X.A random variable X is uniform from 4 to 8. A Gaussian random variable Y has mean of 10. Approximate the variance ofYY if only 2.5% of its elements is a subset of X.
- Suppose that the number of accidents occurring on a highway each day is a Poisson random variable with parameter λ=3.a) Find the probability that 3 or more accidents occur today.b) Repeat part (a) under the assumption that at least 1 accident occurs today.Suppose that the number of accidents occurring on a highway each day is a Poisson random variable with parameter λ=3.1i. Find the probability that 3 or more accidents occur today ii. Find the probability that no accident occur todaySuppose that the number of accidents occurring on a highway each day is a Poisson random variable with parameter (lambda) =3.1 Find the probability that 3 or more accidents occur today Find the probability that no accidents occur today
- The number of times that a person has a cold each year is a Poisson random variable with parameter λ = 5. Suppose that a new drug has been developed that reduces this to λ = 3, but that the drug only works for 75% of the population. For the remaining 25% of the population there is no effect. A person takes the drug for an entire year. During the year the person had 2 colds. What is the probability that the drug had an effect on this person?Suppose that the number of emergency patients in a given day at a certain hospital is a Poisson random variable X with parameter20. What is the probability that in a given day there will be at most 5 emergency patients?X is an poisson random variable with parameter λ = 4 Calculate P {X ≤ 3}
- Suppose the random variable y is a function of several independent random variables, say x1,x2,...,xn. On first order approximation, which of the following is TRUE in general?A single observation of a random variable having a hypergeometric distribution with N = 7 and n = 2 is used to test the null hypothesis k = 2 against the alternative hypothesis k = 4. If the null hypothesis is rejected if and only if the value of the random variable is 2, find the power of the test.Suppose the lifespan (in months) of a smartphone battery can be modeled as a continuous random variable with CDF F(x) = 1 − e-x/3 x ≥ 0 What is the probability that the battery lasts between 12 to 15 months?