4. [8 marks] Suppose that X has a "standard half-normal distribution", meaning that its density function is: fX​(x)=π2​​

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4. [8 marks] Suppose that X has a "standard half-normal distribution", meaning that its density function is: fX​(x)=π2​​e−2x2​,x>0 For parts a) and b), you will need to use Maple syntax to enter your answers. a) Determine the expected value E(X). E(X)= (click on the preview button to check your answer.) b) Derive fY​(y), the density function of Y=2+3X. fY​(y)= Q , for y∈ (click on the preview button to check your answer.) c) You can calculate cumulative probabilities for X from RStudio, using the fact that X=∣Z∣ (where Z is standard normal) and hence P(X≤x)=P(−x≤Z≤x) Use this result and the RStudio to compute P(X≤1). P(X≤1)= Q (Enter your answer correct to 4 decimal places)