STAT 4195-10 FALL 2023 Homework Assignment I
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Homework Assignment I (Due on Thursday, 11/02/2023)
STAT 4195-10
FALL 2023
Please show work for partial credit. Students are expected to abide by the GW code of academic integrity.
1) (6 points) Toss a fair die and two fair coins. (Hint: Use (a) to compute (b) and (c))
(a) Write down the sample space.
(b) Find the probability of getting exactly two heads.
(c) Find the probability that the number on the die is at most 2.
2) (6 points) Suppose a box consists of 4 white, 3 blue and 3 green marbles.
(a) How many ways can you pick 4 marbles out of this box?
(b) Find the probability that there are no white marbles among the 4 marbles.
(c) Find the probability that there is exactly 2 blue and 2 green marbles among the 4 marbles.
3) (2 points) Twenty people have gathered at a party. Find the number of handshakes when each person shakes hands with everyone else at the party.
(4) (4 points) The manager of a local soft drink company believes that when a new
beverage dispensing machine is set to dispense 7 ounces, it in fact dispenses an amount X
at random anywhere between 6.5 and 7.5 ounces. Suppose that X has a uniform distribution.
(a) Find P (X > 7.1)
(b) If you pick five bottles at random, what is the probability that exactly three bottles will contain more than 7.1 ounces?
5) (4 points) Suppose X denotes the normal random variable with mean 5 and variance 4. Compute the following
(a) P (X > 2)
(b) P(2 < X < 7)
(6) (8 points) The heights of adult males are known to be normally distributed with a mean of 70 inches and a standard deviation of 2 inches.
(a) Find the probability that a randomly selected adult male will be taller than 75 inches?
(b) If three adult males are picked at random, find the probability that at least one of the males is taller than 75 inches.
(c) If you want 90% of the adults to pass through a door freely, how high should be the door?
7) (6 points) The survival rate during a risky surgery for patients is 80%. Suppose ten patients undergo this surgery.
(a) Find the probability that at least two people will survive the surgery.
(b) What is the expected value and the standard deviation of the number of patients among these ten patients who will survive the surgery.
(c) Find the probability that the number of people who will survive this surgery exceeds the mean.
8) ( 5 points) Suppose you toss a coin until head shows up where P(H) = p. Let X be the total number of tosses. Find the probability that X is an odd integer.
9) (5 points) Suppose X follows Poisson distribution with mean 5.
Find E(X(X- 1)(X-2))
10) (5 points) Suppose X follows normal distribution with mean 5 and variance 16, find the third moment of X.
2023-11-02