# Essay on Probability Theory and Random Variable

Words: 964
Pages: 4

Event A is rolling a die and getting a 6. Suggest another event (Event B) that would be independent from Event A.
A company runs 3 servers, each providing services to 40 computers. For each server, two of its client computers are infected. What is the probability that 3 randomly chosen client computers serviced by different servers (one per server) will all be infected?
The probability that Alice’s RSA signature on a document is forged is () What is the probability that out of 4 messages sent by Alice to Bob at least one is not forged?
Event A is selecting a “red” card from a standard deck at random. Suggest another event (Event B) that is compatible with Event A. What is the probability of getting 6 tails in 10
Find the math expectation of Ryan’s winnings after 3 games.
Find the variance of Ryan’s winnings for a single game.
Find the standard deviation of Ryan’s winnings for a single game.
Does it pay for Ryan to play this game at the fair? Explain.
Find the cumulative distribution function of Ryan’s winnings for a single game and draw its graph.
(5 points each for parts a-e and 20 points for part f)
X -\$5 \$0 \$10 \$50
P 0.60 0.20 0.10 0.10
Find , , if a random variable is given by its density function, such that(25 points) QUOTE , if QUOTE QUOTE , if QUOTE QUOTE , if QUOTE Let be given by its distribution function , such that QUOTE , if QUOTE (25 points) QUOTE , if QUOTE QUOTE , if QUOTE
Graph the distribution function
Graph the density function
Find ,,
A random variable is distributed normally with QUOTE and QUOTE . Find P(3≤X<4.5). The distribution of the length of a standard piece of computer paper is normal with an expectation of 11 inches and the standard deviation of 0.1 inch.
Find the probability that the length of any given piece of computer paper is between 10.95 and 11.10.
Find the probability that the length of any given piece of computer paper is less than 10.90.
Find the probability that the length of any given piece of computer paper is greater than 11.20.
The density function of a random variable is given by

Find its (a) math expectation, (b) variance and (c)