**Assignment Task**

**Task**

1. The number of medical emergency calls per hour has a Poisson distribution with parameter m. A record of emergency calls is available for a sufficient amount of time and parameter m is assumed to be the same throughout the available recording of calls. Numbers of emergency calls at different hours are considered independent.

If m = 1, find probabilities:

a. That number of calls occurs in 10 consecutive hours of a single medical response team shift will be more than 4 but less than 16?

b. What is the probability of more than 20 emergency calls occur in 10 consecutive hours of a single medical response team shift? (It is considered as impossible to serve more than 20 calls for an emergency team.)

2. Let X1 = N(1,4) and X2 = N(2,1) with correlation ½.

Find

a. P (X1 < X2> b. P (X1 < X2 xss=removed> 3. Die is cast independently N times until a six appears on the up side of the die at a first time. So N is a random variable with possible values 1,2,3,…. Find those probabilities p(n)=P(N=n) for all n= 1,2,3,… and prove that it is a discrete distribution. Find cumulative distribution function for random variable N with discrete mass function probabilities p(n) and sketch it. Find E(N) and Var(N).

Mary and Bob take turns throwing a die, until one of them throws a six to win. Mary starts the game. Do Mary and Bob have an equal probability of winning? Find P(N=2,4,6,8,…), is it equal to probability of Mary win P(N=1,3,5,7,…)?

4. Let probability density of random variable X is f(x)=x/2 for 0 5. Let X1 and X2 has bivariate normal distribution with mean values are 0, variances 1 and correlation ½.

Let Y1 = 3-1/2 ( X1 + X2) and Y2 = X1 – X2 . Find

P( .1 < (Y1 )2 + ( Y2 )2 < 6>

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