Suppose that X has a geometric distribution with parameter p ∈(0,1) (see formula (3.17) on page 129 for the pmf). For any natural number n show that P(X ≤n) = 1 −(1 −p)n+1. (b) Suppose now that Xn has a geometric distribution with parameter λ/n, where λ ∈(0,1). Show that the distribution function of Xn/n converges to the distribution function of an exponential random variable X with parameter λ, that is, for every x > 0, P(Xn n ≤x) −→P(X ≤x). You can pretend as if nx is always a natural number. (c) What does the geometric distribution model in a binomial process? From here, explain what the exponential distribution models in a Poisson process.
Suppose that X has a geometric distribution with parameter p ∈(0,1) (see formula (3.17) on page 129 for the pmf). For any natural number n show that P(X ≤n) = 1 −(1 −p)n+1. (b) Suppose now that Xn has a geometric distribution with parameter λ/n, where λ ∈(0,1). Show that the distribution function of Xn/n converges to the distribution function of an exponential random variable X with parameter λ, that is, for every x > 0, P(Xn n ≤x) −→P(X ≤x). You can pretend as if nx is always a natural number. (c) What does the geometric distribution model in a binomial process? From here, explain what the exponential distribution models in a Poisson process.
Suppose that X has a geometric distribution with parameter p ∈(0,1) (see formula (3.17) on page 129 for the pmf). For any natural number n show that P(X ≤n) = 1 −(1 −p)n+1. (b) Suppose now that Xn has a geometric distribution with parameter λ/n, where λ ∈(0,1). Show that the distribution function of Xn/n converges to the distribution function of an exponential random variable X with parameter λ, that is, for every x > 0, P(Xn n ≤x) −→P(X ≤x). You can pretend as if nx is always a natural number. (c) What does the geometric distribution model in a binomial process? From here, explain what the exponential distribution models in a Poisson process.
Suppose that X has a geometric distribution with parameter p ∈(0,1) (see formula (3.17) on page 129 for the pmf). For any natural number n show that P(X ≤n) = 1 −(1 −p)n+1. (b) Suppose now that Xn has a geometric distribution with parameter λ/n, where λ ∈(0,1). Show that the distribution function of Xn/n converges to the distribution function of an exponential random variable X with parameter λ, that is, for every x > 0, P(Xn n ≤x) −→P(X ≤x). You can pretend as if nx is always a natural number. (c) What does the geometric distribution model in a binomial process? From here, explain what the exponential distribution models in a Poisson process.
Expression, rule, or law that gives the relationship between an independent variable and dependent variable. Some important types of functions are injective function, surjective function, polynomial function, and inverse function.
Expert Solution
Step 1
Sol:-
(a) The probability that X is less than or equal to n is given by:
P(X ≤ n) = P(X = k) = (1-p)(k-1)p
Using the formula for the sum of a geometric series, we have:
