Compute the Probability that a Randomly Chosen Voter - Statistics Assignment Help

Assignment Task:

TASK:

1. Voters living in the city vote for candidate A with probability 0.6, while voters living in the country vote for candidate A with probability 0.4. It is known that 80% of all voters live in the city and the rest 20% live in the country.

(a) Compute the probability that a randomly chosen voter will vote for candidate A. [3]

(b) Compute the probability that a person who will vote for candidate A lives in the city. [3]

(c) Four randomly surveyed voters indicated they would vote for candidate A. Let X be the number of voters among these four who live in the city. Compute the probability P(X = 3). [3] 2. Suppose X1 and X2 are two independent random variables that have the following moment generating function (mgf): M1(t) = E etX1 = M2(t) = E etX2 = 0.5et 1 0.5et , t< ln 2. (a) Compute the probability P(min(X1, X2) > 2). [3]

(b) Define Y = X1 + X2. Compute the probability P(Y = 4). [3]

(c) Compute the probability P(Y > 7). [3] (d) Compute Var(0.5Y + 80). [3]

3. Mutation in a certain gene can occur with probability 0.001 in human population. Suppose X people in a random sample of 2500 people will be observed to have this mutation. Note that X follows a binomial distribution.

(a) Compute P(X ? 2). [2]

(b) A binomial distribution b(n, p) can be approximated by a Poisson( = np) distribution if p is small and n is large. Use this result to approximate the probability in part (a) by a Poisson probability. [2

] (c) The probability in part (a) may also be approximated by a normal probability based on the central limit theorem. Give a normal approximation (using the continuity correction) to P(X ? 2). [2]

4. Let X1 and X2 be two independent Bernoulli(p = 0.5) random variables. Define two new random variables: Y1 = min(X1, X2) and Y2 = max(X1, X2).

(a) Compute the joint probability mass function (pmf) of (Y1, Y2). [3]

(b) Compute E(Y1), E(Y2), Var(Y1) and Var(Y2). [4]

(c) Compute Cov(Y1, Y2). Are Y1 and Y2 independent? Why or why not? [3] Page 2 of 5 5. Let X be a continuous random variable with probability density function (pdf) f(x) = 8 < : c if 4

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