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If P (A) = 1/3
, P (B^c) = 1/4
, then P (AB) = 0.
True or False?
A population consist of 2,4,5,9 and 10, with sample size of 3. Compute the mean and variance of the
sampling distribution of the sample mean.
The total number of hours, measured in units of
100 hours, that a family runs a vacuum cleaner over a
period of one year is a continuous random variable X
that has the density function
f(x) =
⎧
⎨
⎩
x, 0 2 − x, 1 ≤ x < 2,
0, elsewhere.
Find the variance of X.
Find the average number of hours per year that families run their vacuum cleaners.
Toss a fair coin twice. You win Ghc 1 if at least one of the two tosses comes out heads.
(a) Assume that you play this game 300 times. What is, approximately, the probability that you win at least Ghc 250?
(b) Approximately how many times do you need to play so that you win at least Ghc 250 with probability at least 0.99?
Roll a die n times and let X be the number of times you roll 6. Assume that n is large.
(a) Compute the expectation E[X].
(b) Write down an approximation, in terms on n and Φ, of the probability that X differs from its expectation by less than 10%.
(c) How large should n be so that the probability in (b) is larger than 0.99?
How many times do you need to toss a fair coin to get 100 heads with probability 90%?
A roulette wheel has 38 slots: 18 red, 18 black, and 2 green. The ball ends at one of these at random. You are a player who plays a large number of games and makes an even bet of Ghc 1 on red in every game. After n games, what is the probability that you are ahead? Answer this for n = 100 and n = 1000.
Suppose that on average 2 people in a major city die each year from alien attack. Suppose that each attack is random and independent.
(a) If X is the number of deaths from alien attack within the next year from a randomly selected major city, what type of random variable is X?
(b) Use the Poisson approximation to approximate the probability that the next major city you visit will have at least 3 deaths due to alien attack?
(c) Why do you think a Poisson approximation is used instead of a normal approximation?
48000 fair dice are rolled independently. Let X count the number of sixes that appear.
(a) What type of random variable is X?
(b) Write the expression for the probability that between 7500 and 8500 sixes show.
(c) The sum you wrote in part (b) is ridiculous to evaluate. Instead, approximate the value by a normal distribution and evaluate in terms of the distribution Φ(x) = P(N(0,1) ≤ x) of a standard normal random variable.
(d) Why do you think a normal distribution is a good choice for approximation?
Suppose X is normally distributed with mean -3 and variance 0.25. Determine the value of c such that
(a) P(X ≤ c) = 0.8
(b) P(|X| > c) = 0.1
(c) P(−c < X ≤ 1) = 0.5
(d) P(|x + 2|≤ c) = 0.99
(e) P(|x−µ| < c) = 0.95
