We model the following coin toss experiment. Toss a coin n times. Let A1, A2, An that X₂ = 1 if ith coin toss lands on head; otherwise X₂ = 0, i = 1, ..., n. Then each X, is a Bernoulli random variable with distribution function f(X) = px (1 - p)1-X₁, where the unknown parameter p = Pr(X; = 1), i=1,...,n. We aim to make inference of the unknown parameter p, the probability of getting a head, based on the data X₁,..., Xn. a. Find the joint distribution f(X1,..., Xn) of the data X₁,.., Xn. b. Find the log-likelihood function of p based on the result of a. c. Find the maximum likelihood estimator (MLE) of p based on the result of b.

Q: axa-1 f(y|a, 0) 0 0,0 > 0. ga (a) Show that the expected value and variance of Y are a E(Y) = ( 0…

A: Y is a random variable having the following pdf- f(y)=αyα-1θα,0<y<θ, α>0, θ>00,otherwise

Q: Let Y1, Y2,...,Y, denotes a random sample from the uniform distribution on the interval (0,0 + 1).…

A: Given :Let Y1,Y2,..,Yn denotes a random sample from the uniform distributionon the interval…

Q: Let X be the random variable, and create two new copies of this X₁, X₂ ~X x | P(X = x) 0 0.5 1 0.2 2…

A: To determine whether 2X and X1 + X2 have the same distribution, we need to compute their respective…

Q: Exercise 4. Let = {a,b,c,d} and define a probability measure on 22 by P(a) = P(c) = 1/6 and P(b) =…

A: c) To verify that E(X|Y) is a function of Y, we need to show that the value of E(X|Y) only depends…

Q: The random variables X₁, X₂ have the same distribution as shown follows Xi P 0 1/2 1 1/4 2 1/4 Where…

A: Here, we are given that the random variable X1, X2 have the same distribution…

Q: C2. Let X be a random variable. (a) Let YaX be another random variable. What is EY, in terms of μ =…

A:

Q: Suppose §1, $2, ... are iid random variables having mean u and variance o?. Form the random sum SN =…

A: Given that ξ1, ξ2, ..., ξn are iid random variables having mean μ and variance σ2.  SN=ξ1+ξ2+...+ξN

Q: EXERCISE 5 Suppose that X1,·,Xm and Y1, , Yn are two samples, with X; between the sample means, X –…

A: We have to use moment generating function to prove that difference between sample mean of two normal…

Q: For any two random variables Y₁ and Y₂, Var(Y₁Y₂|Y₁ = y₁) = y{Var(Y₂|Y₁ = y₁). TRUE

A: Given that Y1 and Y2 are two random variables. Formula: If X is a random variable, then VX=EX2-EX2.…

Q: let Y be a random variable that can either be equal to 0 to 1. let p(Y=1)=p,what is E(|Y-p|)?

A:

Q: 2. We have n = 100 many random variables X;'s, where the Xi's are independent and identically…

A: It is stated that, the random variables Xi' s are iid Bernoulli random variables, i=1,2, ,.., 100 .…

Q: Suppose x is a normally distributed random variable with u = 31 and o = 6. Find a value xo of the…

A:

Q: Suppose random variable X has a Poisson distribution with t​ = 2 and λ= 4. What is​ E(X)?

A:

Q: N(u = 0, 02 = 4), Y2 ~ N(µ = 3, o? Consider the three normal random variables Y1 Y3 ~ N(u = -3, o² =…

A:

Q: If X is a continuous random variable that takes on values between 10 and 40, then the P(X = 15.5) =…

A: We have given that  X is a continuous random variable that takes on values between 10 and 40, then…

Q: Exercise 13. Let X₁,..., Xn~ N(x, o2) i.i.d. and Y1₁,..., Yn~ N(My, 02) i.i.d. with known o². We…

A: Let X1,X2,...,Xn~Nμx,σ2 iid and Y1,Y2,...,Yn~Nμy,σ2 iid with known σ2. Di=Xi-Yi EDi=μx-μy VarDi=2σ2…

Q: b. Suppose of = 2, o% = 2.5, and m = n. Find the sample size n so that X - Ý will be within one unit…

A: X1,X2,...,Xm and Y1,Y2,...,Yn are two samples with  Xi~ N(µ1 ,σ1)  , Yi~ N(µ2 , σ2)  All the random…

Q: PMF of random variable X is defined as 0.2 X = -2 0.1 X = -1 0.1 X = 0 Px (x) = {0 X = 1 ve X = 2…

A:

Q: Consider the three normal random variables Y Ν(μ-0,σ?4) , Y , Ν (μ Y3 ~ N(µ = -3, o² = 3). Find the…

A:

Q: Exercise 4. Let = {a,b,c,d} and define a probability measure on 222 by P(a) = P(c) = 1/6 and P(b) =…

A: a) To list all sets in σ(Y), we need to consider all possible events that can be formed from Y.…

Q: 4. Suppose X and Y are independent random variables such that E(X)= 0, E(Y)= -1, Var(X)= 1, and V…

A:

Q: q13- X and Y are two independent random variables. We know V(X) = 26.43 and V(Y) = 23.67, then V(8X…

A:

Q: Let A(0) = $100 and A(1) = $110. Also let S(0) = $25, and suppose that the random variable S(1) has…

A: The given problem is solved using the formula for forward contract. The detailed solution is…