4. Let X be a positive random variable (i.e. P(X< 0) = 0. Argue that(a) E(1/X) > 1/E(X)(b) E(-log(X)) > -log(E(X))(c) E(log(1/X))> log(1/E(X))(d) E(X³) > (E(X))³

Jenson's inequality states that if h is continuous and convex function on interval I and X is a…

Answered: Let X be a positive random varia (a)…

Let X be a positive random varia (a) E(1/X) > 1/E(X) (b) E(-log(X)) >-log(E(X)) (c) E(log(1/X)) > log(1/E(X) (d) E(X³) > (E(X))³

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter5: Inverse, Exponential, And Logarithmic Functions

Section: Chapter Questions

Problem 9T

See similar textbooks

Related questions

Q: For x = R, f(x) = log2 - sinx| and g(x)=f(f(x)), then (1) g'(0)=-cos(log2) (2) g is differentiable…

Q: Show E(X)

A: Here given joint distribution of x and y

Q: Please

A: We write f(x) using ln(x) because we know derivative of ln(x) We use the change of base formula of…

Q: If f(x) = log,(cot(x) + a?) , then f'(x) = %3D (A) 1 (cot(x)+2?) (B) csc (x)+2x (cot(x)+r) 1 (C)…

A: By using differentiation formula and change of basis formula, we calculate the derivative of given…

Q: S(x) = log, (4x + 11. Find /"(2).

Q: Find the equation of the secant line for f(x) = log2 (3x² + 2x – 1) from x = 1 to x = 3.

Q: Find E(X)

Q: 3 Differentiate if x* log,(2+vx) f(x)=|

A: Here the differentiation is in the following steps

Q: Suppose that the time in minutes, Y , that an insect spends on a plant is exponentially distributed…

A: Y is the time in minutes that an insect spends on a plant and .

Q: -1)/(x4))1/2

Q: Let g(x) = log3(æ). Find g'(4). g '(4).

Q: Show that b) logi = (2n + 2) ri лi (n = 0, ±1, ±2, ...);

Q: g(x)=log,(5x4-3*). Find g'(x)

Q: Find the exponential function f(x)=a whose graph is given as:

A: From the graph we can see that the value of function at x = -1 is f(-1) = 3 Now we will look the…

Q: log x+ log(x²/y) + log(x³/y²)...... to n terms

Q: If f(x) = ln(4x + ln(x)) find f ′(1)

Q: Show E(XY)

Q: If Y Exponential(8), a > 0 and b > 0, show that P(Y > a+b[Y > a) = e-b/B.

A: P[A|B] = P[A∩B]/P[B] So, P[Y>a+b | Y>a] = P[Y>a+b ∩ Y>a] / P[Y>a] for a>0, b>0,…

Q: Find (x + 2)e3* dx

Q: can you prove E(x^2) = Var(x)+[E(x)]^2 ?

A: Variance:The expected squared difference between a random variable and the mean can be termed as…

Q: Q5) Find (x) a) log,(27) + log,(x) = 3.5 b) 2log,(x) – log,(x + 6) = 1 c) log,(x) + log.(x+ 5) = 2…

A: Given: logx(27) + log9(x) =3.5 2log3(x) - log3(x+6)=1 log6(x) +log6(x+5)=2

Q: Find the exponential function f(x) = ax whose graph is given. f(x) y 4 2, 25 -3 -2 -1 1 2 3 -1 2.

Concept explainers

Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.

Question

4. Let X be a positive random variable (i.e. P(X< 0) = 0. Argue that
(a) E(1/X) > 1/E(X)
(b) E(-log(X)) > -log(E(X))
(c) E(log(1/X))> log(1/E(X))
(d) E(X³) > (E(X))³

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

Step 2

Step 3

Step 4

Step 5

Step by step

Solved in 5 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Point Estimation, Limit Theorems, Approximations, and Bounds$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, statistics and related others by exploring similar questions and additional content below.

Similar questions

Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.
Recall that the general form of a logistic equation for a population is given by P(t)=c1+aebt , such that the initial population at time t=0 is P(0)=P0. Show algebraically that cP(t)P(t)=cP0P0ebt .
Consider the function: f(x)= −1 − x^2 Which of the following is true? A. This function is a pdf only for some values of random variable B. This function cannot be a pdf for any set of values of random variable C. None of these D. This function is a pdf for any set of values of random variable.
1. Fix r € N, and suppose that Y₁,..., Yn are independent and identically distributed NegBin(r, ) random variables, each with probability mass function given by ¡v (u | 0) – (1 + r − ¹) (1 - fy y (1 - 0) ³0⁰ for y ¤ N and unknown 0 € (0, 1). Here the number of successes r in the negative binomial is known.
(1) Let X = b(16,- find E(4-3x) and distribution function.
Suppose the random variable T is the length of life of an object (possibly the lifetime of an electrical component or of a subject given a particular treatment). The hazard function hr(t) associated with the random variable T is defined by hr(t) = lims-o- P(t ≤ T
1. In an epidemic, the probability of having S|D is 0.2 and P(D) = 0.1. P(S|D') = 0.04 and P (D') = 0.01. What is the P(D|S)? 2. The relationship between weight and age was found to have a linear relationship, with expression weight= 3.0 (age) +10. Predict the weight of a girl whose age is 20 years? 3. A. If a z score of 1.95 is equal to a p of 0.9744, what proportion is greater than1.95 B. From question A, what proportion is between the mean and 1.95? 4. If a the constant for a regression is 0.8 and the standard deviation for the x variables is 4 while the standard deviation for the y variables is 6. What is the correlation coefficient r? 5. In a class of 20 students, twelve take mathematics and genetics, while eight take genetics only. What is the probability of selecting a student who takes only mathematics?
Let N(t) be the percentage of a state population infected with a flu virus on week t of an epidemic. What percentage is likely to be infected in week 4 if N(3) = 8 and N'(3) = 1.2?
Suppose that each individual in a large insurance portfolio incurs losses according to an exponential distribution with mean 1/2,where i varies over the portfolio according to a G(a, 3) mixing distribution. The respective densities of the two distributions are given by Sx (x) = (1/2) exp (-x/à), x > 0, 2 > 0; S(2) =- T(a) 2-l exp(-8i), 2 > 0. Given that the Pareto pdf given by fx (x) = (5 +x)ª+1 * > 0, a > 0, 8> 0. (a) Show that the marginal distribution of losses follows a Pareto distribution, i.e. P(a, 5). (b) Use the mixing formulation of the Pareto to deduce that if X~P(a, 5), then E (X) = .
7. Consider the two-class logistic model with the linear predictor: p(x) log (₁ - p(x)) = ³0 + B₁X1 + B₂X2 + ... + BpXp· 1 (a). Write down an expression for the odds O(x): = p(1|x) p(2x) (b). Show that the odds ratio comparing the odds for two new cases with explanatory O(x') variables x and x' does not depend on Bo. (c). What does this imply if we simply want to rank new cases in terms of p(1|x).
LTE OB/s e O VOLTE0 196% 8:54 AM MATHS_LOG... (B) (C) 4 equal O512 Q.3 (D) 03 Q.34 (log, 3) x (log; 625) equals: (В) 2 (A) 1 (C) 3 (D) 4 Q.35 (log, 5) (log, 9) (log; 2) is equal to: 3 (A) I (B) 2 (С)2 (D) 5 ese Q.36 If log, 27 = a, then log, 16 is: 3-a (A) 4(3+ a) 3+a (B) 4(3–a) 4(3 + a) (C) (3-a) 4(3-а) (D) (3+ a) - log 10 32 Q.37 If log, 5+ log10 (5x + 1) = log , (x + 5) + 1, then x is equal to: (A) I (B) 3 (C) 5 (D) 10 2 Q.38 If logs (x² + x) – logs (x + 1) = 2, then the value of x is: (A) 5 (B) 10 (C) 25 (D) 32 Q.39 The value of is: log, 60 log, 60 log, 60 2 (A) 0 (B) I (C) 5 (D) 60 The value of (log, 4) (log, 5) (log, 6) (log, 7) (iog, 8) (log, 9) is: (A) 2 Q.40 X (B) 7 (C) 8 (D) 33 n the value Q.41 The value of 16 og,5 is: 5 (A) 64 (B) 5 (C) 16 (D) 25 If log x + log y = log (x + y), then (A) x = y Q.42 (В) ху 1 (C) y = X-/ (D) y= X-1 I to: 16 a Q.43 If log "+ log = log( a + b), then: a of x is: (A) a + b= 1 (C) a = b (B) a - b = | (D) a - b- | 24 b log ab qual to: Q.44 + log bc…
1. Mean, 2. Variance, 3. Coefficient of Variation, 4. T = [(X, – X,)² + (X3 – X4)²]/(n – 1)(n + 1), 5. T, = (X1 + Cos(X,) + Min(X,,X3) + log(X4) – X)²I/(n² – 1), D 6. T3 = [(X, – X2)² + Exp(-(X3 – X4)²)]/[(x1 – X_)² + (X3 – X4)*], 7. TĄ = [Sd(X,,X,2, X3, X4, X5, X6)? + Median(X,,X2, X3, X4, Xg, X6)*]/[(n – 1) (n + 1), 8. Compare between t e samples. |