Probability and Statistics Exam Review, Exams of Data Analysis & Statistical Methods

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STAT 381 Answers to Review for Exam 2

1. Suppose we draw 2 balls out of an urn with 8 red, 6 blue, and 4 green balls. Let X be the number of red balls we get and Y the number of blue balls. (a) Find the joint pmf of X and Y. Y X = 0 1 2 fY 0 6 / 153 32 / 153 28 / 153 66 / 153 1 24 / 153 48 / 153 0 72 / 153 2 15 / 153 0 0 15 / 153 fX 45 / 153 80 / 153 28 / 153 (b) Find the marginals of X and Y. (c) Find the conditional pmf of fY |x(y|x = 1) Y |X = 1 0 1 2 P (Y |X = 1) 32 / 80 48 / 80 0 (d) Find P (Y = 1|X = 1) = 48/ 80
2. Suppose P (X = x, Y = y) = c(x + y) for x, y = 0, 1 , 2 , 3. (a) What value of c will make this a joint density? c = 1/ 48 (b) What is P (X > Y )? 3/ 8 (c) Find E(X), V (X), cov(X, Y ) Marginal fX (x) is X P (X=^ k=^ k) 3 / 240 5 / 241 7 / 242 9 / 243 E(X) = 23 / 12 , V (X) = 155 / 144 , E(XY ) = 168 / 48 = 7 / 2 , Cov(X, Y ) = − 25 / 144.
3. Suppose X and Y has joint density f (x, y) = c(x + y) for 0 < x, y < 1. (a) What is c? c = 1 (b) What is P (X < 1 /2)? 3/ 8 (c) What is P (X + Y > 1 /2)? 23/ 24 (d) Find E(X), V (X), cov(X, Y ) E(X) = ∫^01 ∫^01 x(x + y)dxdy = 7/ 12 , V (X) = ∫^01 ∫^01 x^2 (x + y)dxdy − (7/12)^2 = 11 / 144 Cov(X, Y ) = E(XY ) − (7/12)^2 = ∫^01 ∫^01 xy(x + y)dxdy − (7/12)^2 = − 1 / 144
4. A chromosome mutation believed to be linked with color blindness is known to occur, on the average, once in every 10,000 births. If 20,000 babies are born this year in a certain city, what is the probability that at least one will develop color blindness? What is the exact probability model that applies here? Binomial with n = 20000, p = 10−^4. P (X ≥ 1) = 0. 8647

5. In a certain published book of 520 pages, 390 typographical errors occur. What is the probability that one page, selected randomly by printer as a sample of her work, will be free from errors? Poisson with λ = 390/520 errors per page. P (X = 0) = e−λ^ = 0. 4724
6. The p.d.f of a random variable X is given by f (x) = ex, for x < 0 (a) Find E(X) = ∫^ −∞∞ xf (x)dx = − 1 (b) Find E(e^3 X/^2 ) = ∫^ −∞∞ e^3 x/^2 f (x)dx = 2/ 5
7. The p.d.f of a random variable X is given by f (x) = kxe−x^2 , for x > 0 (a) Find the constant k. k = 2 (b) Find the distribution function F (x). F (x) =

{ 1 − e−x^2 for x > 0 0 for x ≤ 0 (Note: Don’t forget the domain.) (c) Find P (X > 4) = e−^16

8. The p.d.f of a random variable X is given by f (x) = √cx, for 0 < x < 4

(a) Find the constant c. c = 1/ 4 (b) Find the distribution function F (x).

F (x) = 0 , x < 0 = 12 √x, 0 ≤ x < 4 = 1 , x ≥ 4 (c) Find P (X < 14 ) and P (X > 1). P (X < 14 ) = 1/4, P (X > 1) = 1 − F (1) = 1/ 2 (d) Find the mean E(X) and the variance V (X). E(X) = 4/ 3 , V (X) = 64/ 45

9. The distribution function of a r.v X is given by

F (x) =

{ 1 − (1 + x)e−x^ for x ≥ 0 0 for x < 0 Find (a) P (X < 2) = 1 − 3 e−^2 (b) P (1 < X < 3) = 2e−^1 − 4 e−^3 (c) P (X > 4) = 5e−^4 (d) the p.d.f of X: f (x) = xe−x, x > 0

10. A soft drink machine can be regulated so that it discharges an average of μ ounces per cup. If the ounces of fill are normally distributed with standard deviation equal to 0.3 ounces, give the setting for μ so that 8-ounce cups will overflow only 1% of the time. X has N (μ, 0. 32 ) Given P (X > 8) = 0.01 standardize P (Z > z) = 0.01, so z = 2.33 by Table 3. μ = 8 − 0. 3 ∗ 2 .33 = 7.301 oz.
11. Show that the normal density with mean μ and a standard deviation σ obtains maximum at x = μ. What is the maximum value? Show that it has inflection points at x = μ ± σ. f (x) = √ 21 πσ e−^ (x−μ)

2 2 σ^2 f ′^ = −xσ− 2 μ f Critical number: x = μ f ′′^ = [(x−μ σ) 22 −σ^2 ]f (x) f ′′(μ) < 0 implies f (μ) is absolute max by the 2nd derivative test for Absolute Extrema. f ′′^ = 0 implies (x − μ)^2 = σ^2 which gives inflections points x = μ ± σ

12. If the length of life Y (in unit of years) for a battery has a Weibull distribution with p.d.f f (y) =^3 y

2 4 e

−y^3 / (^4) , y ≥ 0. Find the probability that the battery lasts less that 4 years given that it is now 2 years old. P (X < 4 |X ≥ 2) = P P^ (2(0^ ≤< X <^ X <^ 4)4) = 1 − e−^14.

13. Let X be the number of customers login on a web site per minute. Assume X has a Poisson distribution with a mean of 6 login requests per minute. (a) What is the probability that no one requested to log on this site in the next minute? e−^6 (b) Let W be the time in minutes between the 2nd and 3rd requests. What is the distribution name of W? Exponential What is the expected value of W? mean= 16 min or 10 seconds.
14. If a r.v X satisfies a normal distribution with mean 60 and standard deviation
    20. Find the following results using the Empirical rule (68-95-99.7 rule): (a) P (X ≥ 80) = 16% (b) P (|X − 60 | < 40) = 95% (c) P (|X − 60 | ≥ 20) = 32%

(d) P (X < 20) = 0.015%

15. If a r.v. X satisfies a Normal distribution with mean 2, variance 9 and a r.v. Y satisfies a normal distribution with mean 3, variance 4. Suppose X and Y are independent. (a) What is the mean and variance of X + Y? E(X + Y ) = 5, V (X + Y ) = 13 (b) What is the mean and variance of X − Y? E(X − Y ) = − 1 , V (X − Y ) = V (X) + (−1)^2 V (Y ) = 13 (c) What is the mean and variance of 2X − 3 Y? E(2X − 3 Y ) = − 5 , V (2X − 3 Y ) = 22 V (X) + (−3)^2 V (Y ) = 72
16. If a r.v.X satisfies a Poisson distribution with mean 2 and a r.v. Y satisfies a Poison distribution with mean 3. Suppose X and Y are independent. (a) What are the variances of X and Y? V (X) = 2, V (Y ) = 3 (b) What is the mean and variance of X + Y? E(X + Y ) = 5, V (X + Y ) = 5