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MTH4120: Final Practice 3

Problem 1. If \(X\) is a random variable on the probability space \(\left(\Omega, \mathcal F, \mathbb P\right)\), what type of mathematical object is \(\left\{X \leq 7\right\}\)?
  • (A) Moment generating function
  • (B) Real number
  • (C) Function whose domain is \(\mathcal F\)
  • (D) Differential equation
  • (E) Sigma algebra
  • (F) Element of \(\mathcal F\)
  • (G) Function whose domain is \(\Omega\)
  • (H) Function whose domain is \(\mathbb R\)

Problem 2. 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 3. 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 \(X\) and \(Y\).

Problem 4. 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 5. 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 6. 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?