91Ó°ÊÓ

Determine the constant \(c\) in each of the following so that each \(f(x)\) is a \(\beta\) pdf: (a) \(f(x)=c x(1-x)^{3}, 0

Let \(Y\) have a truncated distribution with pdf \(g(y)=\phi(y) /[\Phi(b)-\Phi(a)]\), for \(a

If a fair coin is tossed at random five independent times, find the conditional probability of five heads given that there are at least four heads.

Let an unbiased die be cast at random seven independent times. Compute the conditional probability that each side appears at least once given that side 1 appears exactly twice.

Let \(X\) have a geometric distribution. Show that $$ P(X \geq k+j \mid X \geq k)=P(X \geq j) $$ where \(k\) and \(j\) are nonnegative integers. Note that we sometimes say in this situation that \(X\) is memoryless.