Please provide proof for the attached question

Applied Practice 1. Safety First through Statistics Suppose the strength of a particular metal beam is given by, S=5+2T2-P 'Where T is a random variable representing the forging temperature and P is a random variable representing purity. Suppose the following statements are true about these random variables: 0 T has a uniform distribution on [0, 2]. a Conditional on a value for T, P is has a normal distribution with mean T/ 2 and standard deviation $712. For example, if T = 1, then, 1 E[PIT = 11: 5m, or[P|T = 1] 2 \\Mar 2 1] = 1. (6 points) Compute the expectation of S