1. In a small town, there are 50 births a year. Assume that the probability that a newborn is a girl is50%. How likely is it that in a given year, there are at least 25 and at most 27 girls born. Calculate this probability
a) exactly.
b) by an approximation with the normal distribution.
2. Consider a biased coin that shows heads in of all cases and tails only in of all cases. The coin is flipped consecutively (and independently) 200 times.
a) What is the probability that tails shows up the first time at the 10th flip?
b) Calculate the probability heads shows up more than 150 times (using a suitable approximation).
3. Assume that the joint probability mass distribution pX,Y of the random variable X and Y is given by
;
.
a) Calculate the marginal probability mass distributions pX and pY .
b) What is the probability mass distribution of the random variable Z = X2 + Y ? 4. Assume that X and Y are jointly distributed random variables with joint density
if 0 ≤ x < ∞, 0 ≤ y ≤ 1;
X,Y ( ) =
0 else.
for some c ∈R.
a) Calculate c.
b) Calculate the marginal probability density functions fX and fY .
c) What is the probability P[3X + Y 2 > 4]?
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