Let X be a normal random variable with = 9 and o 1.85 and Y be a normal random variable with u = 3 and o 0.45. Assume X and Y are independent. Find the following probabilities: (a) P(X < 10, Y < 1.5) (b) P(X> 8,Y < 1.75) (c) P(7.75 ≤X ≤ 10.25, Y> 3.75) (d) P(X < 12, 2.1 ≤Y ≤ 3.8)

4. Let X₁ N (0,3²), X₂ N(4,4²), and X3 ~ N(-1,22) be independent random variables. Find P (X₁ + X₂ +3X3 <-8), using the table for standard normal distribution.

Let Y be a binomial random variable with n = 10 and p = 0.3. (a) P(3 < Y < 5) = P(3 ≤ Y < 5) = (b) P(3 < Y ≤ 5) = P(3 ≤ Y ≤ 5) =

Let random variable X be the number of users communicating with a cellular base station within a given time interval. Its PMF is :

The random variable X takes values -1, 0, 1 with probabilities 1/8, 3/8, 4/8 respectively. a) Write the CDF of X. b) Write the PMF of Y = X² + 2. %3D c) Compute E(Y).

Suppose that X follows a Binomial distribution, i.e., X ∼ Binomial(3, 0.5). Define a new random variable as Y = X 2 , then, what is the value of the PMF of Y evaluated at 1, i.e., what is Pr(Y=1)? a). 0.3125 b). 0.25 c). 0.375 d). 0.5

3. Let X be the random variable that takes on the integers {0, 1, 2, ..., 15} with equal probabilities. Define a new random variable Y = X + A, where A is a random variable that takes on the values {-1, 0, 1} with equal probabilities. If the RVs X and A are independent, find the mutual information between X and Y.

Let X be a normal random variable with  = 10 and  = 2. Find the values of P(X < 11) b. P(X > 8) P(7 < X < 9)

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

If X is a normal random variable with mean μ = 3 and standard deviation σ = a. P(X > 11) b. P(2-3|X > -5) 4, find