Single variable sum probability

Questions asking for the probability of a sum of independent observations of a single normal variable (e.g. X₁ + X₂ + ... + Xₙ where all Xᵢ have the same distribution).

2 questions

CAIE S2 2017 June Q7
7
  1. A random variable \(X\) is normally distributed with mean 4.2 and standard deviation 1.1. Find the probability that the sum of two randomly chosen values of \(X\) is greater than 10 .
  2. Each candidate's overall score for an essay is calculated as follows. The mark for creativity is denoted by \(C\), the penalty mark for spelling errors is denoted by \(S\) and the overall score is defined by \(C - \frac { 1 } { 2 } S\). The variables \(C\) and \(S\) are independent and have distributions \(\mathrm { N } ( 29,105 )\) and \(\mathrm { N } ( 17,15 )\) respectively. Find the proportion of candidates receiving a negative overall score.
OCR S3 2013 June Q1
1 The blood-test procedure at a clinic is that a person arrives, takes a numbered ticket and waits for that number to be called. The waiting times between the numbers called have independent normal distributions with mean 3.5 minutes and standard deviation 0.9 minutes. My ticket is number 39 and as I take my ticket number 1 is being called, so that I have to wait for 38 numbers to be called. Find the probability that I will have to wait between 120 minutes and 140 minutes.