Page 1 of 2 CSS 415: Methods of Mathematical Statistics I Fall 2024 Homework No. 03

1. Question: The probability distribution of random variable, X, is defined as follows: X 0 1 2 3 4 Probability 0 0.3 0.1 0.3 0.3 (a) Explain if the above is a valid probability model. (b) Is the described random variable discrete or continuous? (c) What is the expected value (mean) of the probability distribution? (d) Calculate P(X = 0). (e) Calculate P(X = 3). (f) Calculate P(X < 4). (g) Calculate P(X > 0). (h) Calculate P(X = 5). (i) What is the total area under any density curve?
2. Question: (Textbook Section 2.4, Q6 on page 112) The element titanium has five stable occurring isotopes, differing from each other in the number of neutrons an atom contains. If X is the number of neutrons in a randomly chosen titanium atom, the probability mass function of X is given as follows: x 24 25 26 27 28 p(x) 0.0825 0.0744 0.7372 0.0541 0.0518 Tab. 1: Probability mass function of the number of neutrons in a titanium atom. (a) Find μ(X) or E(X), the expected value of X. (b) Find σ(X), the standard deviation of X.
3. Question: (Textbook Section 2.4, Q22 on page 115) The concentration of a reactant is a random variable X with probability density function (PDF) given by: f (x) =    2e−2x 1−e−2 0 < x < 1 0 otherwise (a) What is the probability that the concentration is greater than 0.5? (b) Find the mean concentration. (c) Find the probability that the concentration is within ±0.1 of the mean. (d) Find the standard deviation σ of the concentrations. (e) Find the cumulative distribution function (CDF) of the concentration.
4. Question: (Textbook Section 2.5, Q10 on page 125) A gas station earns $2.60 for each gallon of regular gas it sells, $2.75 for each gallon of midgrade gas, and $2.90 for each gallon of premium gas. Let X1, X2, and X3 denote the numbers of gallons of regular, midgrade, and premium gasoline sold in a day, respectively. Assume that X1, X2, and X3 have means μ1 = 1500, μ2 = 500, and μ3 = 300, and standard deviations σ1 = 180, σ2 = 90, and σ3 = 40, respectively. (a) Find the mean daily revenue. (b) Assuming X1, X2, and X3 to be independent, find the standard deviation of the daily revenue.
5. Question: (Textbook Section 2.5, Q16 on page 126) The thickness X of a wooden shim (in mm) has the probability density function f (x) =    3 4 − 3(x−5)2 4 4 ≤ x ≤ 6 0 otherwise . (a) Find μX. (b) Find σ2 X. (c) Let Y denote the thickness of a shim in inches (1 mm = 0.0394 inches). Find μY and σ2 Y. (d) If three shims are selected independently and stacked one atop another, find the mean and variance of the total thickness.