Show that the following function is a valid probability mass function of the random variable X. p(x) = (()*, x = 1,2,3,...
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- b) Suppose that a continuous random variable X is associated with the following function: (i) Show that c = =1/15 2 71 Page 0 < x < 2, Scx, f(x) = {0, elsewhere. lo, (ii) Calculate the mean and variance of X. (iii) Find the CDF of the random variable X, F(x). (iv) Use either the probability distribution function (pdf) of X, f(x), or the cumulative probab distribution function (cdf) of X, F(x), to compute P(0.25 < x < 1.25).9. Consider the formula: P(x) = x/21, x 0, 1, 2, 4. Is P(x) a probability mass function? If so, show the distribution of X in tabular form and compute the expected value of X.a) Consider the following probability density distribution of random variable X. if 0a. Explain the difference between the method of moment generating function and the method of incarnation to find the probability density function of a random variable b. Let X1, X2, X3 ,,,,, each of them has Poisson distribution with parameter lamda 1, lamda2, lamda3,........ i. Get the probability distribution of Y=X1+X2+....Xn ii. Get the mean and variance of Y c. Y1 and Y2 be a joint probability function as shown below in the screenshot imageU=Y1/(Y1+Y2), Using the method of incarnation (Jacobian), find the probability density function of UConsider a random variable X taking the values k1, k2, , km E R .. with probability ; Pn E [0, 1] 1. Write down the formula for the P1, P2, .. respectively, where p1 + P2 + expected value of f(X) for a given function f(-). + Pn ...Two discrete random variables X and Y have joint probability mass function (pmf) (a) (b) (c) ƒ(x) = { 5 0 Calculate the value of k. Show that f(x|y) Show that f(y x) = k n(n+1) = 1 n 1 8 x = 1,2,..., n; y=1,2,...,x. otherwiseI need help finding the probability that only two cars arrive.Is P(S = s) = fs(s) = 1+11+1 ; s = 0,1,2,... 0; е. w. a discrete probability function? Why or why not?5. Suppose that X is a discrete random variable with probability density function p(x) = cx², x = 1, 2, 3, 4. (c) Find Var(X). Select one: O a. 18.4 O b. 354 O c. none O d. 342.81. Verify whether the following functions can be considered as probability mass functions: (i) P(x = x) = x² + 1 18 (iii) P(X= x) = -, x = 0, 1, 2, 3 (ii) P(X - x) - ²-2, -, x= 1, 2, 3 2x + 1 18 -, x = 0, 1, 2, 3 [Ans.: Yes) [Ans.: No] [Ans.: No]Let X be a random variable with the probability mass function (PMF) 0.05, =-2 0.25, г — 0 = 1 p(x) = 0.25, г — 2 %3| 0.35, x = 0.1, * = 5 otherwise 0, We can also summarize the key information of the above PME into a table; P(X = =) -2 0.05 0. 0.25 1. 0.25 0.35 0.1 What is the probability that X is non-negative and less than 2? O 0.5 O 0.1 O 0.25 O 0.85 2.For the continuous random variable X, the probability distribution is given by a(x? + 1) f(x) = { 1sx<4 otherwise (a) Find i. The value of the constant a. ii. E(X). iii. Var(X) iv. P(1 sXs 3)
