Let X have a Poisson distribution with parameter λ. Show that E(X)=λ directly from the definition of expected value. (Hint: The first term in the sum equals 0, and then x can be canceled. Now factor out λ and show that what is left sums to 1.)

Let X have a Poisson distribution with parameter λ. Show that E(X)=λ directly from the definition of expected value. (Hint: The first term in the sum equals 0, and then x can be canceled. Now factor out λ and show that what is left sums to 1.)

Name: Let X have a Poisson distribution with parameter λ. Show that E(X)=λ d
Uploaded: 2024-03-20T07:06:41.000Z
Channel: Bartleby
Description: Let X have a Poisson distribution with parameter λ. Show that E(X)=λ directly from the definition of expected value. (Hint: The first term in the sum equals 0, and then x can be canceled. Now factor out λ and show that what is left sums to 1.)

Algebra and Trigonometry (MindTap Course List)

4th Edition

ISBN:9781305071742

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter14: Counting And Probability

Section14.2: Probability

Problem 3E: The conditional probability of E given that F occurs is P(EF)=___________. So in rolling a die the...

See similar textbooks

Similar questions

The conditional probability of E given that F occurs is P(EF)=___________. So in rolling a die the conditional probability of the event E, getting a six, given that the event F, getting an even number, has occurred is P(EF)=___________.
The conditional probability of E given that F occur is P(EF)= _____________. So in rolling a die the conditional probability of the event E. “getting a six,” given that the event F, “getting an even number.” has occurred is P(EF)= ____________.
Given that the discrete random variables X has probability mass function, Pr(X=x)= (x/6 where x=1,2,3) 0 elsewhere, Describe and graph its cumulative distribution function,F(x)
Let X be an exponential random variable with standard deviation σ. FindP(|X − E(X)| > kσ ) for k = 2, 3, 4, and compare the results to the boundsfrom Chebyshev’s inequality.
Suppose that you have ten lightbulbs, that the lifetime ofeach is independent of all the other lifetimes, and that eachlifetime has an exponential distribution with parameter l.a. What is the probability that all ten bulbs fail beforetime t?b. What is the probability that exactly k of the ten bulbsfail before time t?c. Suppose that nine of the bulbs have lifetimes that areexponentially distributed with parameter l and thatthe remaining bulb has a lifetime that is exponentiallydistributed with parameter u (it is made byanother manufacturer). What is the probability thatexactly five of the ten bulbs fail before time t?
Suppose a quality control expert examines iterms for defects in a series of independent fixates, each of a fixed duration, and suppose that defects are present. Let p be the probability that the defect or flaw is detected and 1-p the probability that a defect or flaw is not detected. Let the r.v. X = the number of defects detected in n fixations. a. What is the pmf of X? b. What is the expected value of X = E(X)? c. What is the moment generating function (mgf) for the r.v. X (Remember to state the interval of validity for t)? d. Use it to find the Variance of X.
Assume that interest rate is 0.02, growth rate of the Geometric Brownian Motion of the stock price is 0.03, and its volatility is 0.1. The current stock price is $100. If one independent sample from a standard normal distribution is 0.6, simulate the stock prices at t = 0.05 using the Euler-Maruyama scheme under the risk-neutral probability measure.
A masking tape manufacturer expects 0.04 flaws per meter of tape, on average. The Poisson assumptions hold. Find the probability of exactly 1 flaw in 0.1 meters of tape. 0.0138 0.0289 0.0103 0.0221 0.0040
Verify that if T has a t distribution with ν degrees offreedom, then X = T2 has an F distribution with ν1 = 1and ν2 = ν degrees of freedom.
Let X denote the voltage at the output of a microphone, and suppose that X has a uniform distribution on the interval from −1 to 1. The voltage is processed by a "hard limiter" with cutoff values −0.5 and 0.5, so the limiter output is a random variable Y related to X by Y = X if |X| ≤ 0.5, Y = 0.5 if X > 0.5, and Y = −0.5 if X < −0.5. (a) What is P(Y = 0.5)? (b) Obtain the cumulative distribution function of Y. Please show your work, thanks!
Consider the distribution of Binomial, Poisson, Exponential, Gamma, and let T1=X+4 and T2=(X-1) Check the property of biasdness