4.Suppose that X is a random variable for which E(X)=µ and Var(X)=o². Show that E(X(X-1)]=µ(µ-1)+o?
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![4.Suppose that X is a random variable for which E(X)=µ and Var(X)=o².
Show that
E(X(X-1)]=µ(µ-1)+o?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9b1ac93b-2030-49ee-bbfa-587197c07896%2Ff6d16412-aab6-43d6-a39b-95ac77ab22a8%2F9c5v8ig_processed.png&w=3840&q=75)
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- Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.Consider a random variable X with E[X] = 10, and X being positive. Estimate E[ln√X] using Jensen’s inequality.If X is an exponential random variables with rate 1, then its distribution function is given by F(x) = 1 − e−x Show that x = − ln(1 − u).
- X is an exponential random variable with λ =1 and Y is a uniform random variable defined on (0, 2). If X and Y are independent, find the PDF of Z = X-Y2A 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).?Consider a random variable X with E[X] = 10, and X being positive. Estimate E[ln square root(X)] using Jensen’s inequality.
- Suppose that n observations are chosen at random from a continuous pdf fY(y). What is the probability that the last observation recorded will be the smallest number in the sample? I asked this question earlier today, but didn't quite understand all of the response. P(y1<=yn)p(y2<=yn) and so on was used, but shouldn't the yn be listed first in the inequality since we want to know if yn is the smallest?Find the variance by calculating the first two moments of the random variable X = (- 1 / λ) ln (1-U), where U ~ U (0,1) and λ> 0.If X1, X2, ... , Xn constitute a random sample from anormal population with μ = 0, show that ni=1X2inis an unbiased estimator of σ2.
- Consider a function F (x ) = 0, if x < 0 F (x ) = 1 − e^(−x) , if x ≥ 0 Is the corresponding random variable continuous?If X is a continuous random variable with X ∼ Uniform([0, 2]), what is E[X^3]?Suppose that three random variables X1, X2, X3 form a random sample from the uniform distribution on interval [0, 1]. Determine the value of E[(X1-2X2+X3)2]