3) The probability density function for Y is given by -7₁ e y > 0 (0, elsewhere f(y) = 30 And its cumulative distribution function is given by F(y) = {1-e%, y > 0 0, elsewhere a) Use the cumulative distribution method or transformation method to show that is a pivotal quantity. (6)
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- (26) Let X b(8,- Var-x) and distribution function.and variance of X if the probability mass function is f(x) = ; ;)" , Find the moment generating function, mean x = 1, 2, 3, 4, ...Suppose that the continuous random variable X has cumulative distribution function given by F(x)= 0, if x < sqrt(2); x2-2, if sqrt(2) <= x <= sqrt(3); 1, if sqrt(3) <= x Find the probability density function of X.
- A variable X is distributed at random between the values 0 and 4 and its probability density function is given by: f(x) = kx³ (4- x)². Find the value of k, the mean and standard deviation of the distribution.4. A continuous random Variable X has probability Density function defined by f(x) = 5-5x; 0<=x<=1, =0 otherwise, find the mean and the variance of the distribution.18.) As a measure of intelligence, mice are timed when going through a maze to reach a reward of food. The time (in seconds) required for any mouse is a random variable Y with a probability density function given by fy(y) = y² -& y ≥ b elsewhere where b is the minimum possible time needed to traverse the maze. What is the cumulative distribution function for Y?
- The probability density function of X, the lifetime of a certain type of electronic device (measured in hours), is given by fx(x) = S if æ > 5 if æ < 5 (a) what is the value of C? (b) What is the cumulative distribution function of X?Suppose that the random variable X has the probability density function for -1s x <1 c(1- x2) f(x) elsewhere What is the variance of X A 1/3 B 1/5 (c) 1/2 D 1/4Suppose X and Y have the joint probability distribution P(X = x, Y = y) = c/x+y| for x = -1,0,2 and y = -1,1 (c is a constant), please answer the following questions. (a) What is the value of c? (b) What is the (marginal) probability distribution of X? (c) What is the (marginal) probability distribution of Y?