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MTH4120: Final Practice 2

Problem 1. A fair coin is tossed \(28\) times. Denote by \(X\) the number of heads in the first \(18\) tosses and by \(Y\) the number of heads in the last \(22\) tosses. Calculate the expected value \(\mathbb E\left[X\cdot Y\right]\).

Problem 2. Assume that \(T\) is an exponential random variable with parameter \(\sigma\). If \(\theta\) and \(\rho\) are two real numbers, which of the following is the correct formula for the expected value \(\mathbb E\left[\left(\rho+\theta T\right)^2\right]\)?
  • (A) \( \int_{0}^{+\infty} \rho^2e^{-\sigma \rho}\,d\rho+ \int_{0}^{+\infty} \theta^2e^{-\sigma \theta}\,d\theta\)
  • (B) \(\frac{2\theta^2}{\sigma^2}+\frac{2 \theta \rho}{\sigma}+\rho^2\)
  • (C) \(\frac1{\sqrt{2\pi}}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} \left(\theta T+\rho\right)^2e^{-\frac12\left(\frac{\theta-\sigma}{\ln \sigma}+\frac{\rho-\sigma}{\ln \sigma}\right)^2}\,d\theta d \rho\)
  • (D) \(\frac1{\ln \sigma\sqrt{2\pi}}\int_{-\infty}^{+\infty}\frac{z^2}{\theta z+\rho}e^{-\frac12\left(\frac{z-\sigma}{\ln \sigma}\right)^2}\,dz\)
  • (E) \(\frac13\theta^2\left(\sigma^2+\sigma\ln \sigma+\ln \sigma^2\right)+\theta\rho\left(\sigma+\ln \sigma\right)+\rho^2\)
  • (F) \(2 \theta \rho \sigma \ln \sigma\)
  • (G) \(\int_{\sigma}^{\ln \sigma}\int_{\sigma}^{\ln \sigma}\left(\theta T+\rho\right)^2\,d\theta d \rho\)
  • (H) \(\left(\theta \ln \sigma\right)^2+\left(\theta \sigma+\rho\right)^2\)

Problem 3. Let \(D\) be the region in the plane inside the square with vertices \((5,5)\), \((15,5)\), \((15,15)\), and \((5,15)\) consisting of points that are outside the circle with center \((7,7)\) and radius \(2\sqrt2\). It is known that \(g:\mathbb R^2\to \mathbb R\) is a positive function of two variables and that the joint density function of random variables \(X\) and \(Y\) satisfies \[f_{X,Y}(x,y)=\left\{\begin{array}{ll} g(x,y), & \mbox{ if }(x,y)\in D,\\ 0,& \mbox{ otherwise.}\end{array}\right.\] Which of the following formulas should be used to evaluate \(\mathbb P\left(X \geq Y\right)\)?
  • (A) \(\int_{5}^{11}\int_{13+\sqrt{26y-y^2-161} }^{15}g(x,y)\,dxdy+\int_{11}^{15}\int_{y}^{15}g(x,y)\,dxdy\)
  • (B) \(\int_{5}^{9}\int_{7+\sqrt{14x-x^2-41 } }^{15}g(x,y)\,dydx+\int_{9}^{15}\int_{x}^{15}g(x,y)\,dydx\)
  • (C) \(\int_{5}^{9}\int_{5}^{y}g(x,y)\,dxdy+\int_{9}^{15}\int_{5}^{7-\sqrt{14y-y^2-41}}g(x,y)\,dxdy\)
  • (D) \(\int_{5}^{11}\int_{13+\sqrt{26x-x^2-161} }^{15}g(x,y)\,dydx+\int_{11}^{15}\int_{x}^{15}g(x,y)\,dydx\)
  • (E) \(\int_{5}^{11}\int_{5}^{y}g(x,y)\,dxdy+\int_{11}^{15}\int_{5}^{13-\sqrt{26y-y^2-161} }g(x,y)\,dxdy\)
  • (F) \(\int_{5}^{9}\int_{7+\sqrt{ 14y-y^2-41 } }^{15}g(x,y)\,dxdy+\int_{9}^{15}\int_{y}^{15}g(x,y)\,dxdy\)
  • (G) \( \int_{5}^{9}\int_{5}^{x}g(x,y)\,dydx+\int_{9}^{15}\int_{5}^{7-\sqrt{ 14x-x^2-41} }g(x,y)\,dydx\)
  • (H) \( \int_{5}^{11}\int_{5}^{x}g(x,y)\,dydx+\int_{11}^{15}\int_{5}^{13-\sqrt{ 26x-x^2-161} }g(x,y)\,dydx\)

Problem 4. The random variable \(X\) has a normal distribution. It is known that for every real number \(t\) the following holds \[\mathbb E\left[e^{tX}\right]=e^{4t+ 18t^2}.\] Evaluate \(\mathbb P\left(X\leq 52\right)\).
(A) \(\Phi(5)\) \(\quad\quad\) (B) \(\Phi(6)\) \(\quad\quad\) (C) \(\Phi(7)\) \(\quad\quad\) (D) \(\Phi(8)\) \(\quad\quad\) (E) \(\Phi(9)\)

Problem 5. Determine the constant \(C\) such that the function \[f(x,y)=Ce^{-(7x+12y)}1_{0 < x < y < +\infty}\] is a joint probability density function of two random variables.

Problem 6. Construct an example of two random variables \(X\) and \(Y\) that each have uniform distribution on the interval \([0,1]\) and whose correlation \(\rho\) is strictly bigger than \(0\) and strictly smaller than \(1\), i.e. \(0< \rho< 1\).

Problem 7.

Let \((X,Y)\) be a random vector characterized as follows. The marginal density \(f_X(x) = 2x\cdot 1_{[0,1]}(x)\) and the conditional distribution of \(Y\) given \(X=x\) is uniformly distributed in \([0,x]\). Determine the marginal density \(f_Y(y)\).

Hint: Find the joint pdf \(f(x,y)\) and its support first.

Problem 8. The random variables \(X\) and \(Y\) have joint bivariate normal distribution. Their expectations satisfy \(\mathbb E\left[X\right]=5\) and \(\mathbb E\left[Y\right]=2\). Their variances satisfy \(\mbox{var}(X)=60\) and \(\mbox{var}(Y)=81\). The covariance between \(X\) and \(Y \) is \(63\). Calculate \(\mathbb E\left[\left.X\right|Y=29\right]\).

Problem 9. What is the probability that two uniform random points in the square are such that the center of the square lies inside the circle formed by taking the two points as diameter?