Problem 1-5 1. If X has distribution function F, what is the distribution function of e*?...
Problem 3. 3. For a nonnegative integer-valued random variable X show that i-0 4. A coin comes up heads with probability p. It is flipped until two consecutive heads or two consecutive tails occur. Find the expected number of flips 5. Suppose that PX a)p, P[Xb-p, a b. Show that (X-b)/(a-b) is a Bernoulli variable, and find its variance 3. For a nonnegative integer-valued random variable X show that i-0 4. A coin comes up heads with probability p. It...
The Bernoulli random variable takes values 0 and 1, and has probability function f(x) = px (1 − p)1−x(a) By calculating f(0) and f(1), give a practical example of a Bernoulli experiment, and a Bernoulli random variable. (b) Calculate the mean and variance of the Bernoulli random variable.
A mixed random variable X has the cumulative distribution function e+1 (a) Find the probability density function. (b) Find P(0< X < 1).
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)
Exercise 3.37. Suppose random variable X has a cumulative distribution function F(x) = 1+r) 720 x < 0. (a) Find the probability density function of X. (b) Calculate P{2 < X <3}. (c) Calculate E[(1 + x){e-2X].
-l0 1- e-2x x MO 2) The distribution function for a random variable X is f(x) x <0 Find a) the density function 2 b) the probability that X 4 c) the probability that -3 <x 6inotion
6. Let X be a continuous random variable whose probability density function is: 0, x <0, x20.5 Find the median un the mode. 7. Let X be a continuous random variable whose cumulative distribution function is: F(x) = 0.1x, ja 0S$s10, Find 1) the densitv function of random variable U-12-X. 0, ja x<0, I, ja x>10.
12. The total number of heads for a coin flipped four times is a random variable X with the following probability distribution P(X-0) 0.0625 PX-1) 0.2500 P(X-2) 0.3750 POX-3) 0.2500 P(X-4) 0.0625 Draw a graph of the density function. 13. The total number of heads for a coin flipped four times is a random variable X with the following probability distribution. P(X-0) 0.10 P(X-1) 0.40 P(X-2) 0.20 P(X-3) 0.10 P(X-4) 0.20 Determine the mean and variance of x.
12. (15 points) Let X be a continuous random variable with cumulative distribution function **- F() = 0, <a Inx, a < x <b 1, b<a (a) Find the values of a and b so that F(x) is the distribution function of a continuous random variable. (b) Find P(X > 2). (c) Find the probability density function f(x) for X. (d) Find E(X)