Given that X is a random variable having a Poisson(2) distribution, compute the following: (a) P(X= 7) when λ = 1.5 P(X= 7) = (b) P(X ≤6) when λ = 0.5 P(X ≤ 6) = (c) P(X> 4) when λ = 4 P(X > 4) = (d) P(X < 8) when λ = 0.5 P(X < 8) =

Suppose X is a random variable of uniform distribution between 1 and 7. Find E(X)

For E (0,1) let Xp be a Geometric random variable with parameter p. (a) Find a value of p so that P(Xp> 2.5) = 9. (b) Let An be the event that Xp is even. Determine P(A,) in terms of p. (c) Suppose Y, is a random variable which is equal to the remainder after integer division of X, by 3. Let p = i, and determine the conditional probability mass function of conditioned on the event Yı = 1.

For a random variable, its hazard function also referred to as the instantaneous failure rate is defined as the instantaneous risk (conditional probabilty) that an event of interest will happen in a narrow span of time duration. For a discrete random variable X, its hazard function is defined by the formula hX(k) =P(X=k+ 1|X > k) =pX(k)1−FX(k). For a Poisson distribution with λ= 4.2, find hX(k) and use R to plot the hazard function.