Tay-Sachs disease is a rare fatal genetic disease occurring chiefly in children,

especially of Jewish or Slavic extraction. Suppose that we limit ourselves to families which have 

(a) exactly three children, and 

(b) which have both parents carrying the Tay-Sachs disease. For such parents,

each child has independent probability of getting the disease. Write X to be the random variable

representing the number of children that will have the disease. 

(a)Show that the probability distribution for X is as follows: 

A six-sided die has four green and two red faces and is balanced so that each face 

is equally likely to come up. The die will be rolled several times.

Suppose that we score 4 if the die is rolled and comes up green, and 1 if it comes up red.

Define the random variable X to be this score.

Write  down  the  distribution  of  probability  for  X  and  calculate  the expectation and variance for X.

For two standard dice all 36 outcomes of a throw are equally likely.

Find  for all j and calculate  Confirm that 

Calculation practice for the binomial distribution.

Find P(X = 2),  P(X < 2), P(X > 2) when (a) n = 4, p = 0.2;   (b) n = 8, p = 0.1;(c) n = 16, p = 0.05; (d) n = 64, p = 0.0125.

A wholesaler supplies products to 10 retail stores, each of which will independently make an order on a given day

with chance 0.35. What is the probability of getting exactly 2 orders? What is the probability of getting exactly 3 orders?

Find the expected number of orders per day.

Suppose  that  0.3%  of  bolts  made  by  a  machine  are  defective,

the defectives  occurring  at  random  during  production.  If  the  bolts  are packaged in boxes of 100,

what is the Poisson approximation that a given box will contain x defectives? Suppose you buy 8 boxes of bolts.

What is the distribution of the number of boxes with no defective bolts?

What is the expected number of boxes with no defective bolts? 

Events which occur randomly at rate r are counted over a time period of length s so the event count X is Poisson.

Find P(X = 2), P(X < 2) and P(X > 2) when (a) r = 0.8, s = 1;(b) r = 0.1, s = 8;(c) r = 0.01, s = 200; (d) r = 0.05, s = 200.

Given  that  0.04%  of  vehicles  break  down  when  driving  through  a certain tunnel find the probability of (a) no (b)

at least two breakdowns in  an  hour  when  2,000  vehicles  enter  the  tunnel.  (use  Poisson approximation) 

A process for putting chocolate chips into cookies is random and the number of choc chips

in a cookie has a Poisson distribution with mean 𝜆. Find an expression for the probability that a cookie contains

less than 3 choc chips. 

Use generating functions to find the distribution of  X + Y ,

where X and Y are independent random variables distributed Binomial and Binomial

respectively, for the case where  and  are arbitrary positive integers (they may or may not be equal)

but  In the alternative case, where  but  , find the mean and variance of X + Y . Is the distribution Binomial?

X and Y are independent random variables having Poisson distributions with parameters 𝜆 and 𝛽 respectively.

By using probability generating functions,  prove  that  X+Y  has  a  Poisson  distribution  and  give  its parameter.