Let X ~ Binomial(3,0.5) and Y ~ Poisson(2) The two random variables are independent. Calculate the quantity, Var(2X-5Y)

The spinner is spun one time. Find the probabilities. P(A) = P(C) = %3D A A E P(BUC)= P(B©) = Sample Space: P(B U A)= P(B U A)C = %3D P(D^)= P(AN C)= Page B.

What is the distribution of X + Y when X and Y are independent random variables with uniform distribution in (0, 1)?

The random variable X ( N (u,5^2)and P(X < 23)= 0.9192. (a)  Find the value of u . (b)  Find the value of P(u< X < 23).

Let X be a random variable with mean μ and variance _2. Show that E[(X − b)2], as a function of b, is minimized when b = μ.

Suppose the distribution of some random variable X is modeled as Negative binomial with parameters, p and r, where p is the probability of success and r is the total number of success over the trials. Find the method of moment (MM) estimators for (p,r), in closed algebraic form expressions based on a random sample of size n, X1, X2,...,Xn.

2 (a) Let Y be the random variable that counts the number of sixes which occur when a die is tossed 10 times. What type of random variable is Y? What is P (Y = 3), expected number of sixes,Var(Y )? (b) Let U be the random variable that counts the number of accidents that occur at an intersection in one week. What type of random variable is U? Suppose that, on average, 2 accidents occur per week. Find P(U = 2), E(U), and V ar(U). (c) Let X be the recorded body temperature of a healthy adult in degrees Fahrenheit. What type of rv is X? Estimate its mean and standard deviation, based on your knowledge of body temperatures.

In the first hour of customers using the app they served 180 customers. STACO decided to treat this hour as 60 non-overlapping intervals of one minute each. They then assumed they could model the number of customers served each minute by independent and identically distributed Poisson Random Variables. They called the number of customers served in the ith minute X₁, so X₁ + X₂ + ... + X60 180, X₁~ Poisson(X), and X; and X, are independent if i j. - STACO used this to compute an estimated 95% confidence interval and then told Coffee Kart that they could claim to serve an average of three customers per minute. = i Recall that the maximum likelihood estimator of A is A X. Using the same assumptions as STACO and assuming that n is large enough to apply the Central Limit Theorem, calculate an estimated 95% confidence interval for X. ii Do you agree with the assumptions made by STACO. Justify your answer.

The number of offspring of an organism is a discrete random variable withmean μ and variance σ^2. Each of its offspring reproduces in the same manner. Find the expected number of offspring in the third generation and its variance.

For each random variable defined, describe the set of possible values for the variable, and state whether the variable is discrete or continuous.  (a) U = number of times a surfer has to paddle in front of a wave before catching one (b) X = length of a randomly selected angelfish