Problem set 7: Continuous Random Variables. Normal Distribution

During the seminars, we will work on problems of three different difficulty levels. Category 1 (green) contains introductory problems, Category 2 (blue) contains standard problems, and Category 3 (orange) contains the most challenging problems.

1 Category 1

Let \(X\) be a random variable with uniform distribution on the interval \((-2,3)\). Find \(\mathbb{P}(X>2\mid X>1)\).

Consider a normally distributed random variable \(Z\sim N(0,1)\).

  1. What is \(P(Z>1.2)\)?

  2. What is \(P(-1.24\le Z\le 1.86)\)?

  3. The probability is \(0.70\) that \(Z\) is less than what number?

  4. The probability is \(0.60\) that \(Z\) is greater than what number?

Consider a normally distributed random variable \(X\sim N(5,4)\).

  1. What is \(P(X>6.4)\)?

  2. What is \(P(5.8\le X<7.0)\)?

  3. The probability is \(0.05\) that \(X\) is in the symmetric interval about the mean between which two numbers?

Consider a normally distributed random variable \(X\sim N(6,25)\). Find the following probabilities:

  1. \(P(6<X<6.4)\)

  2. \(P(0\le X<8)\)

  3. \(P(-2<X\le 0)\)

The manufacturer of a brand new lithium battery claims that the mean life of a battery is \(3800\) hours with a standard deviation of \(250\) hours.

  1. What percentage of batteries will last for more than \(3500\) hours?

  2. What percentage of batteries will last for more than \(4000\) hours?

  3. Batteries which will last for more than \(c\) hours constitute more than \(\frac{1}{5}\) of the population. Find the maximum possible \(c\).

Consider a normally distributed random variable \(X\sim N(\mu,\sigma^2)\). It is known that \(P(X\le 66)=0.0359\) and \(P(X\ge 66)=0.1151\).

  1. Give a clearly-labelled sketch to represent these probabilities on a normal curve.

  2. Show that the value of \(\sigma\) is \(5\).

  3. Find \(P(69\le X\le 83)\).

2 Category 2

Random variable \(\xi\) takes values in the interval \((a,b)\), where \(-\infty\le a<b\le\infty\), and has density \(f(x)\). The function \(\varphi(x)\) is strictly monotone and differentiable on \((a,b)\). Moreover \(\varphi'(x)\neq0\) in \((a,b)\).

Find the density of the random variable \(\eta=\varphi(\xi)\). Find the PDF of \(\sqrt{\xi}\), if \(\xi\) has an exponential distribution with parameter \(\lambda\).

Let \(X\) be a random variable with PDF given by

\[ f_X(x)= \begin{cases} c x^2, & |x|\le 1,\\ 0, & \text{otherwise}. \end{cases} \]

  1. Find the constant \(c\).

  2. Find the CDF of r.v. \(X\).

The p.d.f. of \(Y\) is \(g(y)=d\cdot y^{-4}\) for \(1<y<\infty\).

  1. Find \(d\).

  2. Find the c.d.f.

  3. Find \(m\) such that \(\mathbb{P}(Y>m)=0.5\).

3 Category 3

The probability density function (PDF) of a distribution \(\mathbb{P}\), defined on \((\mathbb{R},\mathcal{B}(\mathbb{R}))\), is given by \(p(x)\). Find the corresponding CDF for

  1. \[ p(x)=\frac{\theta}{\pi(\theta^2+(x-x_0)^2)} \]

(Cauchy distribution with scale parameter \(\theta\) and location parameter \(x_0\));

  1. \[ p(x)=k(x-1)^{k-1}I(1\le x\le 2),\qquad k\in\mathbb{N}. \]

A sniper in the shooting range shoots at the ``quarter circle’’, that is, at the area

\[ D=\{(x,y):x^2+y^2<1,\ x>0,\ y>0\}. \]

The distribution of the hitting probability \(\mathbb{P}\) is uniform in the region \(D\). In other words, the density of such a distribution is

\[ p(x,y)=\frac{1}{\pi/4}I((x,y)\in D). \]

Find:

  1. The CDF and the density of the marginal probability distribution \(\mathbb{P}_1\), equal to the projection of \(\mathbb{P}\) along the first coordinate;

  2. The probability of the sniper hitting the square \([0,3/4]\times[0,3/4]\).

Random variable \(\xi\) has standard Cauchy distribution. Find the PDF of

  1. \(\dfrac{\xi^2}{1+\xi^2}\),

  2. \(\dfrac{1}{\xi}\).