If X has the distribution function 0 for x<1 for 1 〈 x < 4 F(x)-〉 for 4 〈 x < 6 for 6a < 10 1 for x 10 f...
Problem 1-5 1. If X has distribution function F, what is the distribution function of e*? 2. What is the density function of eX in terms of the densitv function of X? 3. For a nonnegative integer-valued random variable X show that 4. A heads or two consecutive tails occur. Find the expected number of flips. coin comes up heads with probability p. It is flipped until two consecutive 5. Suppose that PX- a p, P X b 1-p, a...
Random variable X has the following cumulative distribution function: 0 x〈1 0.12 1Sx <2 F(x) 0.40 2 x<5 0.79 5 x<9 1x29 a. Find the probability mass function of X. b. Find E[X] c. Find E[1/(2X+3)] d. Find Var[X]
1) Binomial distribution, f(x) = px (1 – p) n-x , x = 0, 1, 2, …, n n = 10, p = 0.5, find Probabilities a) P(X ≥ 2) b) P(X ≤ 9) 2) f(x) = (2x + 1)/25, x = 0, 1, 2, 3, 4 a) P(X = 4) b) P(X ≥ 2) c) P(X ≥ -3) 3) Z has std normal distribution, find z a) P(-1.24 < Z < z) = 0.8 b) P(-z < Z <...
The random variable X has probability density function f (x) = k(−x²+5x−4) 1 ≤ x ≤ 4 or =0 1 Show that k = 2/9 Find 2 E(X), 3 the mode of X, 4 the cumulative distribution function F(X) for all x. 5 Evaluate P(X ≤ 2.5). 6 Deduce the value of the median and comment on the shape of the distribution.
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.
The random variable X has the following probability distribution function- f(x) = 6x(1-x) when 0 < x < 1 and 0 elsewhere. Find P(X < 0.8 | X > 0.6).
3. Suppose that the distribution function of X is given by b <0 0b<1 1 b<2 2 b<3 3 < b 0 F(b) = (a) Find P{X = i}, i = 1,2,3 (b) Find P< X <} (c) Find the probability mass function of the random variable X-2.
4. The Uniform (0,20) distribution has probability density function if 0 x 20 f (x) 20 0, otherwise, , where 0 > 0. Let X,i,.., X, be a random sample from this distribution. Not cavered 2011 (a) [6 marks] Find-4MM, the nethod of -moment estimator for θ for θ? If not, construct-an unbiased'estimator forg based on b) 8 marks Let X(n) n unbia estimator MM. CMM inbiase ( = max(X,, , Xn). Let 0- be another estimator of θ. 18θ...
Consider the following probability distribution. X 0 2 4 6 P(X = x) 1/4 1/4 1/4 1/4 3. (5 points) Suppose we draw n random samples (X1, ... , Xn), and an estimator 0(X1, ... , Xn) is proposed as n B(x,,,X,) X;I(Xi 70, and X; #6), n i=1 where I(-) is an indicator function, I(X; # 0, and Xi # 6) = 0, if X; € {0,6}, and I(X; # 0, and X; + 6) = 1, if Xi...
1. A random variable X has the cumulative distribution function exe F(X) = 1 + ex • Find the probability density function • Find P(0 < X < 1)