Statistics and Probability Quiz
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Questions and Answers

What is the probability that a standard normal variable takes on a value between -2.3 and 0.14?

  • 0.7449
  • 0.6449
  • 0.5449 (correct)
  • 0.4449

    • What is the memoryless property of exponential distribution?

  • P (Y > s + t|Y > t) = P (Y > s + t); s, t ≥ 0
  • P (Y > s + t|Y > t) = P (Y > t); s, t ≥ 0
  • P (Y > s|Y > t) = P (Y > s); s, t ≥ 0
  • P (Y > s + t|Y > t) = P (Y > s); s, t ≥ 0 (correct)

    • What is the approximate probability that a normal random variable Y takes on a value less than or equal to -7, given a moment-generating function m(t) = e^(-5t+6t^2)?

  • 0.282 (correct)
  • 0.211
  • 0.233
  • 0.104

    • What is the marginal density for Y2, given the joint density p(y1, y2)?

    <p>p(y2) 0.5 0.1 0.1 3 0.5 0.5 y2 - 1 - 2 0.5</p> Signup and view all the answers

    What is the value of F(3, -1), given the joint probability density f(y1, y2) = 4, 0 ≤ y1 ≤ 1, 0 ≤ y2 ≤ 1?

    <p>0.2</p> Signup and view all the answers

    What is the marginal density for Y1, given the joint probability density f(y1, y2) = 4, 0 ≤ y1 ≤ 1, 0 ≤ y2 ≤ 1?

    <p>3(1 - y1)/4</p> Signup and view all the answers

    What is the expression for the marginal density of Y1?

    <p>f1(y1) = 34y1 + 58</p> Signup and view all the answers

    What is the value of Cov(Y1, Y2) given that E(Y1 Y2) = 3/40?

    <p>1/40</p> Signup and view all the answers

    What is the conditional density of Y1 given Y2?

    <p>f(y1|y2) = 14(3y1 + 5y2) / (83 + 54y2)</p> Signup and view all the answers

    What is the condition for Y1 and Y2 to be independent?

    <p>f(y1,y2) = f1(y1)f2(y2)</p> Signup and view all the answers

    What is the moment generating function of a random variable X?

    <p>MX(t) = E(e^(tX))</p> Signup and view all the answers

    What is the value of E(Y1Y2) for the given joint density f(y1,y2) = 6(1-y2) for 0 <= y1 <= y2 <= 1?

    <p>3/20</p> Signup and view all the answers

    What is the moment generating function of a Bernoulli random variable X with probability p?

    <p>MX(t) = pe^(t) + (1 - p)</p> Signup and view all the answers

    What is the moment generating function of a binomial random variable X with parameters n and p?

    <p>MX(t) = (pe^t + (1 - p))^n</p> Signup and view all the answers

    What is the expression for the marginal density of Y2?

    <p>f2(y2) = 83 + 54y2</p> Signup and view all the answers

    What is the value of the conditional density f(y1|y2) at y1 = 0.5 and y2 = 0.7?

    <p>14(3<em>0.5 + 5</em>0.7) / (83 + 54*0.7)</p> Signup and view all the answers

    What is the moment generating function of a Poisson random variable X with parameter λ?

    <p>MX(t) = e^(λ(et - 1))</p> Signup and view all the answers

    What is the moment generating function of a normal random variable X with mean µ and variance σ²?

    <p>MX(t) = e^(µt + σ²t²/2)</p> Signup and view all the answers

    What is the conditional mean of Y given X = x in a linear function form?

    <p>µy|x = µy + ρxy / (σy * σx) * (x - µx)</p> Signup and view all the answers

    If X and Y are independent random variables, what is the value of Cov(X, Y)?

    <p>Cov(X, Y) = 0</p> Signup and view all the answers

    What is the formula for the variance of the sum of X and Y?

    <p>Var(X + Y) = Var(X) + Var(Y) + 2Cov(X, Y)</p> Signup and view all the answers

    What is the value of P(x + y ≤ 1) given the joint probability density function f(x, y) = 4(1-xy) for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1?

    <p>11/18</p> Signup and view all the answers

    What is the correlation coefficient, ρxy, for the joint probability density function f(x, y) = 4(1-xy) for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1?

    <p>-1/13</p> Signup and view all the answers

    What is the conditional mean of Y given X = x, in terms of the mean and standard deviation of Y, and the correlation coefficient ρxy?

    <p>µy|x = µy + (ρxy * σy) / σx * (x - µx)</p> Signup and view all the answers

    What is the moment generating function (MGF) of an exponential random variable X with parameter λ?

    <p>MX (t) = λ λ−t, for t &lt; λ</p> Signup and view all the answers

    What is the joint probability distribution involving two random variables called?

    <p>Bivariate distribution</p> Signup and view all the answers

    If X and Y are independent random variables, what is the relationship between their joint probability distribution and their individual probability distributions?

    <p>f (x, y) = f (x) × f (y)</p> Signup and view all the answers

    What is the notation for the conditional probability distribution of Y given X = x?

    <p>f (y|X = x)</p> Signup and view all the answers

    What is the covariance of X and Y in terms of their means and expectations?

    <p>Cov(x, y) = E(xy) + µx µy</p> Signup and view all the answers

    What is the range of the correlation coefficient, ρxy?

    <p>-1 and 1</p> Signup and view all the answers

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