Suppose T1 and T2 are iid Exp(1)
a)What is the probability density function of T1 + T2?
b)What is the probability that T1 + T2 ≥ 3?
Suppose T1 and T2 are iid Exp(1) a)What is the probability density function of T1 +...
Let Xi, , X. .., Exp(β) be IID. Let Y max(Xi, , h} Find the probability density function of Y. İlint: Y < y if and only if XS for i 1,,n.
If T1 and T2 are independent exponential random variables, find the density function of R=T(2)-T(1). From Mathematical Statistics and Data Analysis by Rice, 3rd Edition, Question 79 from Chapter 3. The solution in the back of the book just says "Exponential (λ)".
Suppose that X1, X2, ..., Xn is an iid sample from the probability density function (pdf) given by where β > 0 is unknown and m is a known constant larger than 1. (a) Show that T-T(X)-Σ-i Xi is a complete and sufficient statistic for Ux(z|β) : β 〉 0} (b) Show that c) For t > 0, show that the conditional density of Xı, given T- t, is 「( mn 1_21) m(n-1)-1 (d) Show that m- 1 mn -...
3 Minimum of IID exponentials Let Z1, ..., Zn be IID exponential random variables with mean 8. That is, each Zi has a PDF given by: f(3) = exp(-2/8], where z and B are positive. Derive the probability density function for min(21, ..., Zn) (i.e., the minimum of random variables 21, ..., Zn). You should find that the probability density function for min(Z1, ..., Zn) is that of an expo- nential random variable. What is the mean of min(Z1, ...,...
Suppose that X has the probability density function f(x) = { 2x 0 < x < 1 0 otherwise Which of the following is the moment generating function of X? 2 et t 2 et t2 2 t2 O t2 2 eet t 2 ett t2 t e eut-1 t
3 Minimum of IID exponentials Let Z1, ..., Zn be IID exponential random variables with mean 8. That is, each Z has a PDF given by: f(3) = exp(-z/B], where 2 and 3 are positive. x f(x) dx Derive the probability density function for min(Z......) (.e., the minimum of random variables 21,..., 2n). You should find that the probability density function for min(Z1,..., Zn) is that of an expo nential random variable. What is the mean of min(21,..., 2..)?
Suppose that X1, X2, ..., Xn is an iid sample, each with probability p of being distributed as uniform over (-1/2,1/2) and with probability 1 - p of being distributed as uniform over (a) Find the cumulative distribution function (cdf) and the probability density function (pdf) of X1 (b) Find the maximum likelihood estimator (MLE) of p. c) Find another estimator of p using the method of moments (MOM)
2te-t2 { 2te-1 = t> 0 6. Let g(t) be the probability density function of the continuous 0 t < 0 random variable X. a. Verify that g(t) is indeed a probability density function. [8] b. Find the median of X, i.e. the number m such that P(X <m) = } = 0.5. [7]
3. Given the survival function: S(t) exp(-t7) derive the probability density function and the hazard function 4 Derive λ t f (t) S(t using the definition of the hazard function and basic definition of conditional probability. 5. Derive S(t) e-) using the definition of the hazard function. 6. Given the hazard function: derive the survival function and the probability density function 7. Prove that if T' has an arbitrary continuous distribution, the cumulative hazard of T, A(T), has an exponential...
Let Xi,..., Xn iid from random variables with probability density function, (0+1)x" 1, ?>0 (x)o for 0 < otherwise (a) Find the method of moments estimator for ? (b) Find the mle for (e): Under which condition is the mle valid?