2. Let X be a Bemoulli random variable. The probability mass function is f(p) p(1 p when x 0 or x 1, where p is the parameter to be estimated. Please declare the MLE, and workout the steps to so...
3. Let X be a continuous random variable with probability density function ax2 + bx f(0) = -{ { for 0 < x <1 otherwise 0 where a and b are constants. If E(X) = 0.75, find a, b, and Var(X). 4. Show that an exponential random variable is memoryless. That is, if X is exponential with parameter > 0, then P(X > s+t | X > s) = P(X > t) for s,t> 0 Hint: see example 5.1 in...
7. Let X1,....Xn random sample from a Bernoulli distribution with parameter p. A random variable X with Bernoulli distribution has a probability mass function (pmf) of with E(X) = p and Var(X) = p(1-p). (a) Find the method of moments (MOM) estimator of p. (b) Find a sufficient statistic for p. (Hint: Be careful when you write the joint pmf. Don't forget to sum the whole power of each term, that is, for the second term you will have (1...
1. Let X be a continuous random variable with probability density function f(x) = { if x > 2 otherwise 0 Check that f(-x) is indeed a probability density function. Find P(X > 5) and E[X]. 2. Let X be a continuous random variable with probability density function f(x) = = { SE otherwise where c is a constant. Find c, and E[X].
2. Let X be an exponentially distributed random variable with parameter 1 = 2. Determine P(X > 4). 3. Let X be a continuous random variable that only takes on values in the interval [0, 1]. The cumulative distribution function of X is given by: F(x) = 2x² – x4 for 0 sxsl. (1) (a) How do we know F(x) is a valid cumulative distribution function? (b) Use F(x) to compute P(i sX så)? (c) What is the probability density...
4.8. Let Z be a random variable with the geometric probability mass function where 0 < π < 1. (a) Show that Z has a constant failure rate in the sense that PriZ kZk1 T for k 0, 1,.... (b) Suppose Z' is a discrete random variable whose possible values are 0, 1, and for which Pr(Z'=KZ2k} = 1-π for k 0,1,.... Show that the probability mass function for Z' is p(k).
1. (15 points) Let X be a continuous random variable with probability density function f (x) c(1-), 0 < 1, where c is a constant. i) Find the constant c ii) What is the distribution function of X? ii) Let Y 1x<0.5 Find the conditional expectation E(X|Y). 1. (15 points) Let X be a continuous random variable with probability density function f (x) c(1-), 0
Let X be a random variable with probability density function 2 (r > 1 0 otherwise. (a) Compute F)-P(X ) (the cumulative distribution function) for 1. Note that F(x) 0 for 1 (b) Let u-F(z). Invert F(-) to obtain 2 marks [1 mark 3 marks) F-1 (u), (z as a function of Your function should have:- Input: n - Number of samples to be generated. . Output: x - (xi, x2,, n) A vector x of n values from the...
3. Let X be an exponential random variable with parameter 1 = $ > 0, (s is a constant) and let y be an exponential random variable with parameter 1 = X. (a) Give the conditional probability density function of Y given X = x. (b) Determine ElYX]. (c) Find the probability density function of Y.
Question 2 Let X1,...,X, be iid Geometric random variables with parameter and probability mass function f(T; 7) = (1 - 7)" for 1 = 0,1,2,... and 0 <I<1. We wish to test: HT=0.50 HT70.50 (a) Find the three asymptotic x1) test statistics (Likelihood Ratio, Wald, and Score) for this setting. versus
Let X be the random variable with the geometric distribution with parameter 0 < p < 1. (1) For any integer n ≥ 0, find P(X > n). (2) Show that for any integers m ≥ 0 and n ≥ 0, P(X > n + m|X > m) = P(X > n) (This is called memoryless property since this conditional probability does not depend on m.)