Math: Introduction To Mathematical Statistics. Complete The ✓ Solved

Math Introduction to Mathematical Statistics. Complete the f

Math Introduction to Mathematical Statistics. Complete the following. Show all work.

(1) Suppose X has PDF f(x) = c e^{-x/2}, x >= 0.

a. Find c to make f a valid PDF.

b. Find the CDF for this value of c.

(2) Suppose X has PDF f(x) = 2x for 0 <= x <= 1.

a. Find E[X].

b. Find Var(X).

(3) Consider a class of three students with scores 75, 80, 85. For samples of size n=2, let Xbar_2 denote the sample mean. What is the sampling distribution of Xbar_2?

(4) Consider discrete random variable X with P(X=x)=k x, x=1,2, else 0.

a. Find k to make this a valid PMF.

b. Find E[X].

c. Find Var(X).

(5) Given a histogram of grades for 30 students:

a. What is the sample proportion scoring between 75 and 85?

b. Which measure of center would you use, mean or median, and why?

(6) A chromosome mutation occurs once in every 10,000 births. Suppose 20,000 babies are born.

a. How many expected to have the mutation?

b. Probability exactly 1 baby has the mutation?

c. Probability the 5th baby surveyed is the first with the mutation?

(7) IQ ~ N(100, 15^2).

a. P(85 < X < 115)?

b. % between 80 and 120?

c. % above 130?

d. % at or above 160?

Paper For Above Instructions

Problem 1. We are given f(x)=c e^{-x/2}, x >= 0. For a valid PDF, integrate over the support and set equal to 1 (Casella & Berger, 2002).

Normalization: ∫_{0}^{∞} c e^{-x/2} dx = c ∫_{0}^{∞} e^{-x/2} dx. Let u = x/2, dx = 2 du, so integral = c 2 ∫_{0}^{∞} e^{-u} du = c 2 * 1 = 2c. Set 2c = 1 ⇒ c = 1/2.

CDF: F(x) = ∫_{0}^{x} (1/2) e^{-t/2} dt = [ - e^{-t/2} ]_{0}^{x} = 1 - e^{-x/2}, for x >= 0. For x < 0, F(x)=0. (This is the exponential distribution with mean 2.) (Ross, 2014)

Problem 2. f(x)=2x on [0,1]. This is a Beta(2,1) density. Compute moments by integration.

E[X] = ∫_{0}^{1} x (2x) dx = 2 ∫_{0}^{1} x^{2} dx = 2*(1/3) = 2/3 ≈ 0.6667. (Wackerly, Mendenhall & Scheaffer, 2008)

E[X^{2}] = ∫_{0}^{1} x^{2} (2x) dx = 2 ∫_{0}^{1} x^{3} dx = 2*(1/4)=1/2 = 0.5. Therefore Var(X) = E[X^{2}] - (E[X])^{2} = 1/2 - (2/3)^{2} = 1/2 - 4/9 = (9 - 8)/18 = 1/18 ≈ 0.05556.

Problem 3. Population scores: {75, 80, 85}. Draw all samples of size n=2 without replacement (standard when sampling students). All unordered pairs are equally likely: (75,80), (75,85), (80,85). Compute the sample mean for each pair:

  • (75,80): X̄ = (75+80)/2 = 77.5
  • (75,85): X̄ = 80
  • (80,85): X̄ = 82.5

Since there are 3 equally likely pairs, the sampling distribution of X̄_{2} is discrete with P(X̄=77.5)=P(X̄=80)=P(X̄=82.5)=1/3. One can compute E[X̄] = population mean = (75+80+85)/3 = 80 and Var(X̄) from the distribution above or using finite-population formulas (Särndal, Swensson & Wretman, 1992).

Problem 4. Discrete PMF: P(X=x)=k x for x=1,2.

Normalization: k(1+2)=3k=1 ⇒ k=1/3.

P(1)=1/3, P(2)=2/3. E[X] = 1(1/3) + 2(2/3) = 1/3 + 4/3 = 5/3 ≈ 1.6667. E[X^{2}] = 1^{2}(1/3) + 2^{2}(2/3) = 1/3 + 8/3 = 9/3 = 3. Var(X) = 3 - (5/3)^{2} = 3 - 25/9 = (27-25)/9 = 2/9 ≈ 0.2222. (Ross, 2014)

Problem 5. (a) Without the actual histogram numeric bin counts we cannot compute a numeric sample proportion. In practice, count the number of students with scores between 75 and 85 (inclusive or as defined by the bin edges), call that count m, and compute sample proportion = m / 30. If the histogram were provided, list the bin counts and compute m/30. (b) Choice of center: if the distribution of grades is approximately symmetric and without outliers, the sample mean is an efficient measure of center; if the histogram is skewed or contains outliers, the median is preferred because it is robust to extreme values (Hogg & Tanis, 2010). For typical grade histograms with skew or outliers, choose the median; for well-behaved symmetric grades, choose the mean.

Problem 6. Mutation probability p = 1/10,000 = 0.0001, n = 20,000.

(a) Expected number = n p = 20,000 0.0001 = 2 babies expected. (b) Exactly 1: use Binomial(n,p): P(X=1) = C(20000,1) p (1-p)^{19999} = 20000 0.0001 (1-0.0001)^{19999}. For large n small p with λ = np = 2, Poisson approximation gives P(X=1) ≈ e^{-2} 2^{1} / 1! = 2 e^{-2} ≈ 2 * 0.135335 = 0.27067 (≈ 27.07%) (Le Cam’s theorem; Ross, 2014).

(c) Probability the 5th baby is the first with mutation: geometric model with success prob p independent each trial: P(first success at trial 5) = (1-p)^{4} p ≈ (0.9999)^{4} * 0.0001 ≈ 0.000099996 ≈ 9.9996 × 10^{-5} (≈ 0.01%).

Problem 7. X ~ N(100, 15^{2}). Use standard normal CDF Φ.

(a) P(85 < X < 115) corresponds to z = (85-100)/15 = -1 and (115-100)/15 = 1, so P = Φ(1) - Φ(-1) = 2Φ(1)-1 ≈ 2(0.8413447)-1 = 0.682689 ≈ 68.27% (empirical rule/normal table) (Rice, 2006).

(b) 80 to 120 ⇒ z = ±(20/15)=±1.3333. Φ(1.3333) ≈ 0.9082 ⇒ P ≈ 2*0.9082 -1 = 0.8164 ≈ 81.64%.

(c) Above 130: z = (130-100)/15 = 2 ⇒ P(X > 130) = 1 - Φ(2) ≈ 0.0228 ≈ 2.28%.

(d) At or above 160: z = (160-100)/15 = 4 ⇒ tail ≈ 1 - Φ(4) ≈ 3.167 × 10^{-5} ≈ 0.00317% (very rare). Normal tail values from standard tables or software (Abramowitz & Stegun; see also statistical software). (Casella & Berger, 2002)

These computations use standard properties of expected value, variance, binomial and Poisson approximations, and the Normal distribution (Wackerly et al., 2008; Ross, 2014). Where the histogram was not provided, the procedure to obtain the answer was stated.

References

  • Casella, G., & Berger, R. L. (2002). Statistical Inference (2nd ed.). Duxbury.
  • Ross, S. M. (2014). Introduction to Probability and Statistics for Engineers and Scientists (5th ed.). Academic Press.
  • Wackerly, D., Mendenhall, W., & Scheaffer, R. (2008). Mathematical Statistics with Applications (7th ed.). Cengage Learning.
  • Hogg, R. V., & Tanis, E. A. (2010). Probability and Statistical Inference (8th ed.). Pearson.
  • Rice, J. A. (2006). Mathematical Statistics and Data Analysis (3rd ed.). Cengage Learning.
  • DeGroot, M. H., & Schervish, M. J. (2012). Probability and Statistics (4th ed.). Pearson.
  • Särndal, C.-E., Swensson, B., & Wretman, J. (1992). Model Assisted Survey Sampling. Springer.
  • Abramowitz, M., & Stegun, I. A. (1972). Handbook of Mathematical Functions. Dover Publications.
  • Le Cam, L. (1960). An approximation theorem for the Poisson binomial distribution. Pacific Journal of Mathematics.
  • National Institute of Standards and Technology (NIST) Digital Library of Mathematical Functions. http://dlmf.nist.gov/ (for tables and distribution function values).

Related Assignments