Suppose that f(x) = 1:5x2 for -1 < x < 1 (0 elsewhere). Determine the following
probabilities:
(a) P(X > 0)
(b) P(X > 0:5)
(c) P(-0.5 �< X </=� 0.5)
(f) Determine x such that P(X > x) = 0.05
(g) Find the cdf of X.
could someone just get me started , I don't know where to start! thanks
Sure! Let's go step by step:
(a) To find P(X > 0), we need to determine the area under the probability density function (pdf) f(x) for values of x that are greater than 0. Since f(x) is defined as 1.5x^2 for -1 < x < 1 and 0 elsewhere, we can see that for x > 0, f(x) is also 0 elsewhere, except now it follows the quadratic function.
To calculate this probability, we need to find the area under the curve of f(x) from x = 0 to x = 1. To do this, we integrate the function f(x) over this domain as follows:
P(X > 0) = Integral[0 to 1] (1.5x^2) dx.
(b) To find P(X > 0.5), we follow a similar process to (a), but this time, we need to integrate from x = 0.5 to x = 1, since we are interested in the area under the curve beyond x = 0.5.
P(X > 0.5) = Integral[0.5 to 1] (1.5x^2) dx.
(c) To find P(-0.5 ≤ X ≤ 0.5), we need to calculate the area under the curve of f(x) between -0.5 and 0.5.
P(-0.5 ≤ X ≤ 0.5) = Integral[-0.5 to 0.5] (1.5x^2) dx.
(d) To determine x such that P(X > x) = 0.05, we need to find the value of x at which the cumulative probability is equal to 0.05. In other words, we need to solve the equation:
P(X > x) = 0.05.
(e) To find the cumulative distribution function (CDF) of X, we need to integrate the pdf f(x) from negative infinity to x. The CDF gives us the probability that X takes on a value less than or equal to x.
Let me know which part you would like me to explain further.