A clinical trial tests a method designed to increase the probability of conceiving a girl. In the study 520 babies were​ born, and 286 of them were girls. Use the sample data to construct a 99​% confidence interval estimate of the percentage of girls born. Based on the​ result, does the method appear to be​ effective?

If the joint probability density function of the random variables X and Y is given by f(x, y) = (1/4)(x + 2y) for 0 < x < 2, 0 < y < 1, 0 elsewhere

(a) Find the conditional density of Y given X = x, and use it to evaluate P (X + Y/2 ≥ 1 | X = 1/2)

(b) Find the conditional mean and the conditional variance of Y given X = 1/2

(c) Find the variance of W = X − 2Y + 3

(d) Find the covariance of X and Y, and determine if X and Y are independent

In an examination the probability distribution of scores (X) can be approximated by normal distribution with mean 64.9 and standard deviation 9.4.

Suppose one has to obtain at least 55 to pass the exam. What is the probability that a randomly selected student passed the exam? [Answer to 3 decimal places]

If two students are selected randomly what is the chance that both the students failed? [Answer to 3 decimal places]

If only top 4% students are given an award, then what was the minimum marks required to get the award? [Answer to 1 decimal place]

Find the probability that in 200 tosses of a fair six sided die, a five will be obtained at least 40 times

The probability that a vehicle will change lanes while making a turn is 55%. Suppose a random sample of 7 vehicles are observed making turns at a busy intersection. a) Find the probability that no more than 4 vehicles will change lanes while making the turn. b) Calculate the expected value u of the vehicles that will change lanes while making turn c) Calculate the standard deviation

Consider the following information:

a)         What is the expected return for A? For B?                                         

b)         What is the standard deviation for A? For B?                                    

c)         What is the expected return on a portfolio of A and B that is 30% invested in A and the remainder in B?       

Probability:

Y is a normal random variable with ? = 0 and ?^2 = 1.

SOLVE FOR expectation E[Y] and variance Var(Y).

Suppose X1, X2, . . . are a sequence of iid uniform random variables with probability density function f(x) = 1 for 0 < x < 1, or 0, otherwise.

Let Y be a continuous random variable with probability density function g(y) = 3y² for 0 < y < 1, or 0, otherwise.

We further assume that Y and X1, X2, . . . are independent. Let T = min{n ≥ 1 : X_n < Y }. That is, T is the first time to get a smaller value than Y in the sequence {X_n} ^∞_n=1(from n=1 to infinity).

(a) Find P(T = 2) = P(X₁ ≥ Y, X₂< Y ).

(b) Find E(T).

(c) Find Var(T)

1. Birthday Problem. In a group of 10 students, what is the probability that
a. Nobody has birthday on the same date
b. At least two have same birthday
c. Exactly two have same birthday
d. Exactly three have same birthday
e. Two or three have same birthday
f. At most three have same birthday

Three boys and three girls are to sit in a row. Find the probability that

i. The boys and girls alternate.

ii. The boys and girls sit together.

iii. Two specific girls sit next to one another.

Please provide full working with correct answer and clear explanation