Statistics and Probability Quiz

Play an AI-generated podcast conversation about this lesson

What is the probability that a standard normal variable takes on a value between -2.3 and 0.14?

0.7449

0.6449

0.5449 (correct)

0.4449

What is the memoryless property of exponential distribution?

P (Y > s + t|Y > t) = P (Y > s + t); s, t ≥ 0

P (Y > s + t|Y > t) = P (Y > t); s, t ≥ 0

P (Y > s|Y > t) = P (Y > s); s, t ≥ 0

P (Y > s + t|Y > t) = P (Y > s); s, t ≥ 0 (correct)

What is the approximate probability that a normal random variable Y takes on a value less than or equal to -7, given a moment-generating function m(t) = e^(-5t+6t^2)?

0.282 (correct)

0.211

0.233

0.104

What is the marginal density for Y2, given the joint density p(y1, y2)?

p(y2) 0.5 0.1 0.1 3 0.5 0.5 y2 - 1 - 2 0.5 Signup and view all the answers

What is the value of F(3, -1), given the joint probability density f(y1, y2) = 4, 0 ≤ y1 ≤ 1, 0 ≤ y2 ≤ 1?

0.2 Signup and view all the answers

What is the marginal density for Y1, given the joint probability density f(y1, y2) = 4, 0 ≤ y1 ≤ 1, 0 ≤ y2 ≤ 1?

3(1 - y1)/4 Signup and view all the answers

What is the expression for the marginal density of Y1?

f1(y1) = 34y1 + 58 Signup and view all the answers

What is the value of Cov(Y1, Y2) given that E(Y1 Y2) = 3/40?

1/40 Signup and view all the answers

What is the conditional density of Y1 given Y2?

f(y1|y2) = 14(3y1 + 5y2) / (83 + 54y2) Signup and view all the answers

What is the condition for Y1 and Y2 to be independent?

f(y1,y2) = f1(y1)f2(y2) Signup and view all the answers

What is the moment generating function of a random variable X?

MX(t) = E(e^(tX)) Signup and view all the answers

What is the value of E(Y1Y2) for the given joint density f(y1,y2) = 6(1-y2) for 0 <= y1 <= y2 <= 1?

3/20 Signup and view all the answers

What is the moment generating function of a Bernoulli random variable X with probability p?

MX(t) = pe^(t) + (1 - p) Signup and view all the answers

What is the moment generating function of a binomial random variable X with parameters n and p?

MX(t) = (pe^t + (1 - p))^n Signup and view all the answers

What is the expression for the marginal density of Y2?

f2(y2) = 83 + 54y2 Signup and view all the answers

What is the value of the conditional density f(y1|y2) at y1 = 0.5 and y2 = 0.7?

14(30.5 + 50.7) / (83 + 54*0.7) Signup and view all the answers

What is the moment generating function of a Poisson random variable X with parameter λ?

MX(t) = e^(λ(et - 1)) Signup and view all the answers

What is the moment generating function of a normal random variable X with mean µ and variance σ²?

MX(t) = e^(µt + σ²t²/2) Signup and view all the answers

What is the conditional mean of Y given X = x in a linear function form?

µy|x = µy + ρxy / (σy * σx) * (x - µx) Signup and view all the answers

If X and Y are independent random variables, what is the value of Cov(X, Y)?

Cov(X, Y) = 0 Signup and view all the answers

What is the formula for the variance of the sum of X and Y?

Var(X + Y) = Var(X) + Var(Y) + 2Cov(X, Y) Signup and view all the answers

What is the value of P(x + y ≤ 1) given the joint probability density function f(x, y) = 4(1-xy) for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1?

11/18 Signup and view all the answers

What is the correlation coefficient, ρxy, for the joint probability density function f(x, y) = 4(1-xy) for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1?

-1/13 Signup and view all the answers

What is the conditional mean of Y given X = x, in terms of the mean and standard deviation of Y, and the correlation coefficient ρxy?

µy|x = µy + (ρxy * σy) / σx * (x - µx) Signup and view all the answers

What is the moment generating function (MGF) of an exponential random variable X with parameter λ?

MX (t) = λ λ−t, for t < λ Signup and view all the answers

What is the joint probability distribution involving two random variables called?

Bivariate distribution Signup and view all the answers

If X and Y are independent random variables, what is the relationship between their joint probability distribution and their individual probability distributions?

f (x, y) = f (x) × f (y) Signup and view all the answers