For a homoge nous Poisson proce ss, the intensity non - function is givn by. Act) =t for t in [o,nJond dcts sy for try S-6. Thot is, If SS=2, derrive the distribution of find PCS_652|ss=2).
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- 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)=___________.Let Xi be arandom sample from U(0,1)prove that Xn’ convarges in probability to 0.50Prove that the rv Y=(tan^-1(X1/X2)) and Z=(X1^2+X2^2)^0.5 are independent. No distribution is given.
- Use the moment generating function to solve. Let X1, . . . , Xn be independent random variables, such that Xi ∼ Poiss(λi), for i = 1, . . . , n.Find the distribution of Y = X1 + · · · + Xn.The probability of the closing of the ith relay in the circuits shown is given by p_i. Let p_1 = 0.4, p_2 = 0.1, p_3 = 0.5, p_4 = 0.7, p_5 = 0.6. If all relays function independently. what is the probability that a current flows between A and B for the respective circuit?An random experiment consists of marking a point randomly inside a half -circle with a radius of 1,as shown below. Let’s call this point N. Let X and Y be the random variables that denote thecoordinates such that X is the horizontal coordinate and Y is the vertical coordinate. Answer thefollowing question. Let g(X, Y ) be the area inside a circle that has the point N as its center and touchesthe half-circle at a single point, as shown below, in gray. Please note that the small circle canextend beyond the half circle on the bottom. Write g(X, Y ) and specify its range. Pleaseprovide the solution step by step.
- Let U be a Uniform(0, 1) RV. Suppose X = 2U-1. What is The probability that X > 0?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 X1, . . . , Xn i.i.d. U([θ1, θ2]), i.e., X1, . . . , Xn are independent and follow a uniform distribution on the interval [θ1, θ2] for θ1, θ2 ∈ R and θ1 < θ2. Find an estimator for θ1 and θ2 using the method of moments.
- 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)? (Answer is not 0.) (b) Obtain the cumulative distribution function of Y. (Need the component for -0.5 <= y < 0.5, and we already know that F(y < -0.5) = 0 and F(y >= 0.5) > 1.) Please show your work, thanks!A metal rod of length L and a sphere of radius R are to be joined to form a small automobile part. L and R are normally distributed with the following parameters: E(L) = 2cm, E(R) = 0.5cm, V(:) =0.1^2 cm, V(R)=0.05^2 cm^2. Assume that L and R are independent. What is the probability that the total length of a single assembly is less than 2.90 cm?Let m(t) be the moment generating function of a random variable X. Show that the random variable W = 10X is m(10t). What is the moment generating function of Z = X-5 in terms of m(t)?
