STA219: Probability and Statistics for Engineering Assignment 3 Solution

Description

1. (10 points) Suppose that the average household income in some country is 900 coins, and the
standard deviation is 200 coins. Assuming the normal distribution of incomes:
(1) Compute the proportion of “the middle class”, i.e., whose income is between 600 and 1200
coins. (5 points)
(2) The government of the country decides to issue food stamps to the poorest 3% of households.
Below what income will families receive food stamps? (5 points)
2. (10 points) Let 𝑋~𝑁(𝜇, 𝜎
2
). Suppose that the probability that the quadratic equation 𝑦
2 + 4𝑦 +
𝑋 = 0 has no real roots (i.e., its discriminant is negative) is 0.5, please determine the value of 𝜇.
3. (10 points) A survey shows that the English score (hundred-mark system) of students approximately
follows a normal distribution 𝑁(𝜇, 𝜎
2
) with 𝜇 = 72. If the number of students with more than 96
points accounts for 2.3% of the total students, what is the probability that the score is between 60
and 84 points?
4. (10 points) Suppose that the diameter of a disc follows a uniform distribution on (𝑎, 𝑏), what is the
expected area of this disc?
5. (10 points) Let 𝑍~𝑁(0, 1). Find E(Φ(𝑍)) and Var(Φ(𝑍)), where Φ is the CDF of 𝑍.
6. (10 points) If 𝑋~𝑁(0, 1), please derive the PDF of the following random variables:
(1) 𝑌1 = |𝑋|; (5 points)
(2) 𝑌2 = 2𝑋
2 + 1. (5 points)
7. (15 points) Suppose that random variable 𝑋 follows an exponential distribution with parameter 2.
Show that both 𝑌1 = 𝑒
−2𝑋
and 𝑌2 = 1 − 𝑒
−2𝑋
follow the uniform distribution on (0,1).
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8. (15 points) Suppose the joint PDF of a random vector (𝑋, 𝑌) is given by
𝑓(𝑥, 𝑦) = {
𝑘𝑒
−(3𝑥+4𝑦)
, 0 < 𝑥, 𝑦 < ∞
0, otherwise.
(1) Determine the constant 𝑘; (5 points)
(2) Find the joint CDF 𝐹(𝑥, 𝑦) of (𝑋, 𝑌); (5 points)
(3) Compute 𝑃(𝑋 + 𝑌 ≤ 1). (5 points)
9. (10 points) Suppose the joint PDF of a random vector (𝑋, 𝑌) is given by
𝑓(𝑥, 𝑦) = {
𝑒
−𝑦
, 0 < 𝑥 < 𝑦 < ∞
0, otherwise,
determine 𝑓𝑋(𝑥) and 𝑓𝑌(𝑦), i.e., the marginal PDFs of 𝑋 and 𝑌, respectively.