P(X = x) = 1/10 where x = 1, 2, 3, ... 10
a) P(X > 6) = P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10) = 5/10 = 0.5
b) P(X > 6 or 5 < X < 7) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10) = 6/10 = 0.6
c) P(X = 1 | X < 5) = P(X = 1) / P(X < 5) = (1/10) / (4/10) = 1/4 = 0.25
d) P(X = 6 and X = 8) = 0
e) P(X = 0 or X = 5) = P(X = 0) + P(X = 5) = 2/10 = 0.2
6. Let X follow a discrete uniform distribution from 1 to 10. Find the following probabilities....
Let X be a random variable following a continuous uniform distribution from 0 to 10. Find the conditional probability P(X >3 X < 5.5). Chebyshev's theorem states that the probability that a random variable X has a value at most 3 standard deviations away from the mean is at least 8/9. Given that the probability distribution of X is normally distributed with mean ji and variance o”, find the exact value of P(u – 30 < X < u +30).
2. Let X be a discrete random variable with the following cumulative distribution function 0 0.2 0.5 ェ<2, 2-1<5.7, 5.7-1 6.5, 6.5 <エ<8.5, F(z)= 18.5 エ a) Find the probability mass function of X b) Find the probabilities P(x>5), P(4<X 6x> 5) c) If E(X) = 5.76, find c.
7. If x is a binomial random variable find the following probabilities: a) P(x = 2) n = 10 and p = .40 b) P (x < 5) for n = 15 and p = .60 8. Find pl, oland o for n = 25 and p = .50
4. Suppose X has a discrete uniform distribution: the distribution function of X 5. A random variable Z has the pmf bclow. P (X-х,)-1 , is|2 n. Find 0 Pz(z) 0.20 0.16 0.4 a (1) What is thevalue of a ? (2) What is P(l S Z <3)? (3) What is Fz (1.7)? 6.
5. Let X be a discrete random variable. The following table shows its possible values associated probabilities P(X)( and the f(x) 2/8 3/8 2/8 1/8 (a) Verify that f(x) is a probability mass function. (b) Calculate P(X < 1), P(X 1), and P(X < 0.5 or X >2) (c) Find the cumulative distribution function of X. (d) Compute the mean and the variance of X.
3. Let X X be a random sample from Uniform[0, where > 0 is unknown. (a) Show that = max{X,X is the MLE of 0. (b) Let the CDF of @ be F(-). Find F(t) for any t e R (c) Find the pdf of 0 Hint: Find the distribution function of Z maxX1,X. The first feu steps will be as follous: F2(2) P(Z) P (maxX, x) ) = P (XS2, X X,) Nert use the fact that Xis are...
Let X be a discrete random variable taking integer values 1, 2, ..., 10. It is also known that: P(X < 4) = 0.57, PCX 2 4) = 0.71. Then P(X = 4) = A: 0.14|B: 0.28 |C: 0.45 OD: 0.64|E: 0.73 OF: 0.95 Submit Answer Tries 0/5
Let X and Y be independent random variables. Random variable X has a discrete uniform distribution over the set {1, 3} and Y has a discrete uniform distribution over the set {1, 2, 3}. Let V = X + Y and W = X − Y . (a) Find the PMFs for V and W. (b) Find mV and (c) Find E[V |W >0].
Let X1,..., X., be a independent identically distributed sample from the DISCRETE uniform distribution with probability mass function 1/0 P(X = x) for a € {1,2,...,0} otherwise i.e. O is an unknown parameter and the probability that X takes any integer between 1 and is constant. Suppose we use @ = 2X - 1 to estimate 0. Find the bias and mean-squared error (MSE) of O. (Note: Di-1 32 = 0(0+1)(20 + 1)/6)
Assume a Poisson distribution with-5.8. Find the following probabilities. b.X-1 с. X>1 Compute the mean and standard deviation for the following hypergeometric distributions. a. n-3, N 9, and E-7 b. n- 5, N-8, and E-3 c. n-6, N 14, and E 3 d. n 4, N-9, and E 4