
Exercise 6 (10 marks) Let X represent the positive difference between the scores obtained from two...
3. Two fair dice are thrown. Let X be the smaller of the two numbers obtained (or the common value if the same number is obtained on botih dice). Find the probability mass function of X. Find P(X>3).
Let X be a random variable corresponding to the sum of scores obtained from rolling two dice. Calculate the following probabilities: 1) p(X ≥ 5) 2) p(8≤ X ≤ 10) 3) p(X ≤ 2)
Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed n times. What are the possible values of X?
Let X represent the number of heads subtracts the number of tails obtained when a coin is tossed 3 times, i.e., X = number of heads − number of tails. (a) Find the probability mass function of X (b) Given that X is at least 0, what is the probability that X is at least 2
6. Let the random variables X and Y represent the population of two species/organisms that compete with each other for survival. Suppose that the probability density function p(x, y) of these random variables is proportional to rye-(az+By) (a) What would be an appropriate sample space S for the random variables? Justify your choice of S. (b) Let p(x, y)-Krye-(az-+8v). What should be the proportionality constant K? (c) What is the most probable set of populations (X, Y)? (Hint: The probability...
Let X represent the difference between the number of heads and the number of tails when a coin is tossed 31 times. Then P(X=11) ==
Let X and Y be two independent and identically distributed random variables that take only positive integer values. Their PMF is pX(n)=pY(n)=2−n for every n∈N , where N is the set of positive integers. Fix a t∈N . Find the probability P(min{X,Y}≤t) . Your answer should be a function of t . unanswered Find the probability P(X=Y) . unanswered Find the probability P(X>Y) . Hint: Use your answer to the previous part, and symmetry. unanswered Fix a positive integer k...
Let X equal the larger outcome when a pair of 6-sided dice are rolled.(a) Assuming the two dice are independent, show that the probability function of \(X\) is \(f(x)=\frac{2 x-1}{36} \quad x=1, \ldots, 6\)(b) Confirm that \(f(x)\) is a probability function.(c) Find the mean of \(X\).(d) Can you generalise \(E(X)\) to a pair of fair \(m\) -sided dice?\(\left[\right.\) Hint: recall that \(\sum_{i=1}^{n} i=n(n+1) / 2\) and \(\left.\sum_{i=1}^{n} i^{2}=n(n+1)(2 n+1) / 6\right]\)
5. (a) (6 marks) Let X be a random variable following N(2.4). Let Y be a random variable following N(1.8). Assume X and Y are independent. Let W-min(x.Y). Find P(W 3) (b) (8 marks) The continuous random variables X and Y have the following joint probability density function: 4x 0, otherwise Find the joint probability density function of U and V where U-X+Y and -ky Also draw the support of the joint probability density function of Uand V (o (5...
1. (10 marks) random variable with density r(x). Let g: R - (a) Let X R be a (differentiable) function and let Y = g(X). Write expressions for the following ((ii)-(iv) should be in terms of the density of X (i) The integral f()d (ii) The mean E(X) (ii The probability P(X e (a, b) (iv) The mean E(g(X)) R be a smooth (1 mark (1 mark) (1 mark (1 mark) (b) Let z E R be a constant and...