Assignments

SUBMISSION GUIDELINES

• Submit your homework by sending it as a PDF file attachment to the email address:

   adiky at gradcenter.cuny.edu

• The subject should specify:

   CSC 21700: HW # Student name

• The file name of the attached work should be following:

  LastName-FirstName-HW#-CSC21700.pdf

  • LastName is your last name

  • FirstName is your first name

  • # is an assignment number

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There is 5% penalty if your work does not comply with above guidelines.

Assignment 1

A. We consider events $A$, $B$, and $C$, which can occur in some experiment. Is it true that the probability that only $A$ occurs (and not $B$ or $C$) is equal to $P(A \cup B \cup C) - P(B) - P(C) + P(B \cap C)$?

B. Let $A$ and $B$ be two events. Suppose that $P(A) = 0.3$, $P(B) = 0.5$, and $P(A \cap B) = 0.1$. Find the probability that A or B occurs, but not both.

C. Three events $E$, $F$, and $G$ cannot occur simultaneously. Further, it is known that $P(E \cap F ) = P(F \cap G) = P(E \cap G) = 1/3$. Can you determine $P(E)$?

D. A fair die is thrown twice. A is the event "sum of the throws equals 5", B is "at least one of the throws is a 3".

1. Calculate P(A | B).

2. Are A and B independent events?

E. Calculate

1. Probability $P(A \cup B)$ if it is given that $P(A) = 1/2$ and $P(B | \bar{A}) = 1/4$.

2. Probability $P(B)$ if it is given that $P(A \cup B) = 3/4$ and $P(\bar{A} | \bar{B} ) = 1/2$.

3. Calculate $P(T)$ when $P(B) = 0.1$, $P(T | B)=0.98$ and $P(T | \bar{B}) = 0.05$.

4. Calculate $P(B | T)$ and $P(B | \bar{T})$ if $P(B) = 0.1$, $P(T | B)=0.98$ and $P(T | \bar{B}) = 0.05$.

Solution

Assignment 2

A. An application is running on three servers $X, Y, Z$ with following availability 96%, 98%, 97%. Assume that the servers are independent. Determine the probability mass function of the server availability. Determine expected value and variance of the server availability.

B. The range of the random variable $X$ is $[0, 1, 2, 3, 4, x]$, where $x$ is unknown. If each value is equally likely and the mean of $X$ is 5, determine x.

C. Two random variables $X$ and $Y$ have the joint distribution, $P(0, 0) = 0.2$, $P(0, 2) = 0.2$, $P(1, 1) = 0.2$, $P(2, 0) = 0.3$, $P(2, 2) = 0.1$, and $P(x, y) = 0$ for all other pairs (x, y).

1. Find the probability mass function of $Z = X + Y$.

2. Find the probability mass function of $U = X - Y$.

3. Find the probability mass function of $V = XY$.

D. Show that for a discrete uniform random variable $X$, if each of the values in the range of X is multiplied by the constant $c$, the effect is to multiply the mean of $X$ by $c$ and the variance of $X$ by $c^2$. That is, show that $E[cX] = c E[X]$ and $Var[cX] = c^2 Var[X]$.

E. The number of requests sent to a server is a Poisson random variable with a mean of 6 requests per hour.

1. What is the probability that 5 requests are received in 1 hour?

2. What is the probability that 10 requests are received in 1.5 hours?

3. What is the probability that less than two requests are received in one-half hour?

F. Show that for any random variables $X, Y , Z, W$, and any non-random numbers $a, b, c, d, e, f$.

G. The probability density function $f$ of a continuous random variable $X$ is

Find $c$ and compute cdf function $F(x)$.

H. The time $X$ it takes to reboot a certain system has Gamma distribution with $\mu = 20$ min and $\sigma = 10$ min.

1. Compute parameters of this distribution.

2. What is the probability that it takes less than 15 minutes to reboot this system?

I. A random variable $Z$ follows a standard normal distribution.

1. Prove that pdf of $Z$ is symmetric, i.e. $F(-a) = 1 - F(a)$ for any $a$.

2. Use above property to compute $P(Z \leq -2)$.

Solution