Let X be a continuous random variable defined on the interval [0, 4] with probability density function: p(x) = c(1 + 4x) (a) Find the value of c such that p(x) is a valid probability density function. (b) Find the probability that X is greater than 3. (c) If X is greater than 1, find the probability X is greater than 2. Also find the probability that X is less than some number a, assuming 0 < a < 4?

Let X and Y have the joint probability density functionf(x, y) = 5, for 0 < x < y < 1.(i) Find the marginal probability density function of X and Y.(ii) Find the conditional probability density function of X, given Y = y.(iii) Find the conditional mean of X, given Y = y.(iv) Find the conditional variance of X, given Y = y.

Let X be a random variable that follows the beta distribution. This random variable is continuous and is defined over the interval from 0 to 1. The probability density function is given by                                         whereand are integers, whose values determine the shape of the probability density function. Because X varies between 0 and 1, we can think of X as the probability that some event (say) E occurs or the proportion of times an event occurs in some population. For example, E could denote the event that a critical part in a newly designed car will lead to a catastrophic failure in accidents at high speeds.   The expected value (i.e., mean) of this random variable is []. That is,                                             . The Excel commands for the beta random variable are           =beta.dist(x,,,true,0,1) for the cumulative probability distribution, and           =beta.dist(x,,,false,0,1) for the probability density function. (a) Now, think in Bayesian terms.…