The probability density function fx) of a continuous random variable x is defined by A 5sxS 10 f(x) = %3D 0, otherwise Then the approx value of A is [Upto 2 decimal placesl
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- Suppose the duration (in hours) of a certain valve is a random variableX with density function f(x) = 648x−4 for x > 6, and zero otherwise. What's the time expected life (in hours) of this valve?Suppose that X is a continuous random variable with density function f(x). If f(x)=k for −5≤x≤3 and f(x)=0 otherwise, determine the value of k.Suppose that two-dimensional continuous random variable (X, Y) has joint probability density function given by f(x,y) = 24xy, x is less than equal to 1 and greater than equal to 0, y is less than equal to 1 and greater than equal to 0, x+y is less than equal to 1 and greater than equal to 0. Check that E(Y) = E[E(Y|X)] and V(Y) = E[V(Y|X)] + V[E(Y|X)].
- Suppose a college professor never finishes her lecture before the end of the class period, and always finishes within five minutes after the class period is supposed to end. Let X = time that elapses between the end of the class period and the actual end of the lecture. Suppose the pdf of X is: (image) Find the value of k that makes f(x) a legitimate probability density function, and use that value of k to find the probability that the lecture ends less than 3 minutes after the class period is supposed to end.X and Y are continuous random variables with pdf f(x,y) = 2x for0 ≤x ≤y ≤1, and f(x,y) = 0 otherwise. Find the conditional expectation ofY given X = x.If we consider the function f (x) = x2 (with x = position) as a probability density distribution defined at 0 ⩽ x ⩽ 3. Determine exactly the integral function F(x).
- Show that the standard normal probability density function f(x) has points of inflection when x = ±1.Suppose that the probability density function of the length of computer cables is f (x) = 2x/(32) for x between 0 and 3 meters. Determine the mean of the cable length. Please enter the answer to 2 decimal places.It is known that the length of time X (in hours) it takes for pizza to be delivered by Domino’s Pizzza hasthe probability density function:f(x) = (12(x^2 − x^3) when 0 < x < 1, and 0 otherwise.a) Find the probability that it will take more than 20 minutes for a pizza to be delivered.b) Find the probability that it will take less than 40 minutes for a pizza to be delivered.c) If 10 independent deliveries are made on a particular day, what is the probability that exactly 7 of them took more than 20 minutes.
- Suppose that X and Y are independent and uniformly distributed random variables. Range for X is (−1, 1) and for Y is (0, 1). Define a new random variable U = XY, then find the probability density function of this new random variable.A random variable X has a Pareto distribution if andonly if its probability density is given by f(x) =⎧⎪⎨⎪⎩αxα+1 for x > 10 elsewhere where α > 0. Show that μ r exists only if r < α.The probability density function of the continuous random variable X defined in the set of non-negative real numbers is given as f(x) = 2.exp(-2x). What is the expected value of X?
