Questions in mathematics

Questions in mathematics

[Free] For all problems on this page, suppose you have data X_1, ..., X_n ~ N(0, 1) that is a random sample of identically and independently distributed standard normal random variables. Useful facts: For a standard normal random variable X_1, we have: E[X] = 0, E[X^2] = 1, E[|X|] = sqrt(2/pi) approx 0.8 The i.i.d. assumption Suppose X_1 is an observation for Bob, X_5 is an observation for Alice, X_7 is an observation for Charlie. Using the following facts about Bob: P(-2 < X_1 < 2) approx 0.95 and P(-sqrt(2) <= X_1 <= sqrt(2)) approx 0.75, compute the probability P(-2 < X_5 < 2, -2 < (X_7)^5 < 2) (Enter the probability P(-2 < X_5 < 2, -2 < (X_7)^5 < 2) or if the probability is not determined uniquely, then enter -1.) P(-2 < X_5 < 2, -2 < (X_7)^5 < 2) = ________ Sample mean Consider the sample mean: bar{X_n} = (1/n)(X_1 + X_2 + ... + X_n). What are the mean E[bar{X_n}] and variance Var[bar{X_n}] of bar{X_n}? E[bar{X_n}] = ________ Var[bar{X_n}] = ________ What kind of distribution does bar{X_n} follow? - Gaussian - Student t - Chi square - Gamma - nonparametric Quantiles Consider the quantile Q_{n, alpha} of order 1 - alpha for the random variable bar{X_n}, that is, the number Q_{n, alpha} such that P(bar{X_n} <= Q_{n, alpha}) = 1 - alpha, 0 < alpha < 1. For alpha < 0.5, as the sample size n increases, does the quantile Q_{n, alpha}: - Increase - Decrease - Stay the same - Oscillate

[Free] Select the correct answer. Which of the following represents a constant from the expression given? [tex]$15 x^2+2 x+9$ A. 2 B. 9 C. 15 D. 24

[Free] What is the slope of the line whose equation is $y-4=\frac{5}{2}(x-2)$?

[Free] Write the expression as an exponent: [tex]$6^{15} \cdot 36$[/tex]

[Free] Which polynomial is factored completely? A. [tex]$121 x^2+36 y^2$[/tex] B. [tex]$(4 x+4)(x+1)$[/tex] C. [tex]$2 x(x^2-4)$[/tex] D. [tex]$3 x^4-15 n^3+12 n^2$[/tex]

[Free] Which of the following is the solution of [tex]$5 e^{2 x}-4=11 ?$[/tex] A. [tex]$x=\ln 3$[/tex] B. [tex]$x=\ln 27$[/tex] C. [tex]$x=\frac{\ln 3}{2}$[/tex] D. [tex]$x=\frac{3}{\ln 3}$[/tex]

[Free] The amount of a sample remaining after [tex]$t$[/tex] days is given by the equation [tex]P(t)=A\left(\frac{1}{2}\right)^{\frac{t}{h}}[/tex], where [tex]$A$[/tex] is the initial amount of the sample and [tex]$h$[/tex] is the half-life, in days, of the substance. A scientist has a [tex]$10-\text{mg}$[/tex] sample of a radioactive isotope. The isotope has a half-life of 8 days. After 16 days, how much of the radioactive isotope remains? A. 2.0 mg B. 2.5 mg C. 5.7 mg D. 7.1 mg

[Free] Solve the equation. [tex]5(e^x+2)=30[/tex] [tex]x=[/tex] $\square$ (Round to three decimal places as needed.)

[Free] By rounding to one significant figure, estimate the answers to these questions: a) [tex]\frac{53 \times 17}{42}=142[/tex] b) [tex]\frac{43 \times 19}{41}=\square[/tex]

[Free] A person is standing exactly 36 ft from a telephone pole. There is a [tex]$30^{\circ}$[/tex] angle of elevation from the ground to the top of the pole. What is the height of the pole? A. 12 ft B. [tex]$12 \sqrt{3} ft$[/tex] C. 18 ft D. [tex]$18 \sqrt{2} ft$[/tex]