. If X is a random variable with probability generating function Px(2)ze-1-*), then , then (a)...
Suppose that X is a continuous random variable with density pX(x) = ( Cx(1 − x) if x ∈ [0, 1] 0 if x < 0 or x > 1. (a) Find C so that pX is a probability density function. (b) Find the cumulative distribution of X. (c) Calculate the probability that X ∈ (0.1, 0.9). (d) Calculate the mean and the variance of X. 9.) Suppose that X is a continuous random variable with density C(1x) if E...
2. The random variable, X has the following probability mass function (i) Find the value of the constant c. HINT: It will help to use the identity = (i) Find the cumulative distribution function of X and sketch both the probability mass function and the cumulative distribution function NOTE: Think carefully about the values of r for which you need to define the distribution function. (ii) Calculate P(X 2 50) and PX 2 50 x2 40
9.) Suppose that X is a continuous random variable with density C(1- if [0,1] px(x) ¡f x < 0 or x > 1. (a) Find C so that px is a probability density function. (b) Find the cumulative distribution of X (c) Calculate the probability that X є (0.1,0.9). (d) Calculate the mean and the variance of X
7. Let X a be random variable with probability density function given by -1 < x < 1 fx(x) otherwise (a) Find the mean u and variance o2 of X (b) Derive the moment generating function of X and state the values for which it is defined (c) For the value(s) at which the moment generating function found in part (b) is (are) not defined, what should the moment generating function be defined as? Justify your answer (d) Let X1,...
A5. The probability generating function of a random variable X takes the form 67,) 0.3 2 G(s) 1-07s Use this probability generating function to compute: (a) The mean and variance of X (b) The probability that X takes the value 2.
Suppose that for a random variable X, X, E(X") 2", n generating function and the probability mass function of X Hint: Use (11.2). 1, 2, 3, . . . the moment
Find the probability generating function of a discrete random variable with probability mass function given by pX(k) = qk−1p, k = 1,2,..., where p and q are probabilities such that p + q = 1. We shall see later that this is called the geometric distribution function.
3. Use the probability generating function Px)(s) to find (a) E[X(10)] (b) VarX(10)] (c) P(X(5)-2) . ( 4.2 Probability Generating Functions The probability generating function (PGF) is a useful tool for dealing with discrete random variables taking values 0,1, 2, Its particular strength is that it gives us an easy way of characterizing the distribution of X +Y when X and Y are independent In general it is difficult to find the distribution of a sum using the traditional probability...
Let be a random variable with probability density function f(x) and moment-generating function 1 1 M(t) = =+ = ? 6 . 6 1 + - 1 36 + -e a) Calculate the mean = E(X) of X b) Calculate the variance o? = E(X -w' and the standard deviation of X
5. Find the moment generating function of the continuous random variable X whose a. probability density is given by )-3 or 36 0 elsewhere find the values of μ and σ2. b, Let X have an exponential distribution with a mean of θ = 15 . Compute a. 6. P(10 < X <20); b. P(X>20), c. P(X>30X > 10), the variance and the moment generating function of x. d.